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Najnowsze preprinty (120)

Ostatnie 120 prac z kategorii math.GT o węzłach, splotach, warkoczach lub plątaninach (tangles), od najnowszej. Przy każdej jest rozwijane streszczenie.

• 21 paź Sang Youl Lee. Affine $G$-multibiquandle modules for colored links I: representability of coloring functors and color-profile polynomials. arXiv PDF

  Let $R$ be a commutative ring with identity, let $G$ be an abelian group, and fix $u\in R[G]^\times$. For each component-labeled oriented link diagram $(D,σ)$, we construct a finitely presented $R[G]$-module $M_u(D,σ;R[G])$ with a universal affine coloring $U_{D,σ}$. Every affine coloring with values in an $R[G]$-module factors uniquely through $U_{D,σ}$ by a module homomorphism. Thus the module represents the affine coloring functor. Each specified colored Reidemeister move induces a unique module isomorphism compatible with coloring transport, and the module isomorphism class is a colored-link invariant. The representing modules and universal colorings commute with arbitrary scalar extension, without a flatness assumption. For classical links, simultaneous rescaling of the component labels removes the common deformation unit. For any multibiquandle, associated-biquandle colorings decompose into fibers indexed by component labelings. For finite targets and palettes, we define a color-profile polynomial recording the joint distribution of fiber sizes and numbers of palette labels used, and prove its invariance under oriented link isotopy. In the affine case, the representing modules determine these fibers, and their specialized dimensions determine the coloring counts over finite fields. The color-profile polynomial is not determined, in general, by the full unweighted distribution of coloring-fiber sizes. It therefore retains information beyond the associated-biquandle coloring count.

• 20 paź Chloé Postel-Vinay. The geometric combinatorics of $k$-shuffle braid groups. arXiv PDF

  This paper is the first in a series of two which aims at introducing a new family of groups called $k$-shuffle braid groups, denoted $\mathcal{B}_{n}(k)$. These groups have two intrinsically related stories. They were first observed by Calegari in arXiv:2105.11265 by looking at spaces of complex monic polynomials of degree $n$ whose critical values lie in $\mathbb{C} \setminus \{1, e^{\frac{2πi}{k}}, \dots, e^{\frac{2π(k-1)}{k}}\}$. The $k$-shuffle braid groups are the fundamental groups of those spaces, which correspond geometrically to complements of $k$ translates of the discriminant variety. The second story starts from the poset of $k$-shuffle partitions, which were introduced by Drew Armstrong and explored in depth in his thesis arXiv:math/0611106. In this paper, we focus on the combinatorial aspects of $k$-shuffle braid groups starting from the poset of $k$-shuffle non-crossing partitions. We build a CW-complex on which the groups act freely, properly discontinuously and cocompactly, which we show is CAT(0) for dimension up to 7. For higher dimension, we show that the CAT(0) property is implied by the conjecture of Brady and McCammond that the dual braid complex is CAT(0) (arXiv:0909.4778).

• 20 paź Ya. Kononov; A. Morozov. Towards differential expansion for generic representations. arXiv PDF

  We report the first results of phenomenological analysis, based on extended set of explicitly calculated colored HOMFLY-PT polynomials in \citeknotebook. They concern differential expansion (DE), which separates the dependencies on the knot ${\cal K}$ and the representation $R$. %of colored HOMFLY-PT polynomials. DE is well understood for symmetric and rectangular representations, while for non-rectangular $R$ even the knot-independent basis $Z_R^X$ of DE is not known. We concentrate on two issues: the possible derivation of DE and on improved characteristic of its defect. We discuss how evolution families – twist knots, double braids and 3-pretzels – can be used to find DE by a peculiar triangular transformation of evolution coefficients. Once $Z^R_X$ are found for particular family, they are expected to stay basically the same for arbitrary knots – modulo controllable reductions, characterized by the defect of DE, which puts knots into different universality classes. Contrary to original expectation, these classes appear labeled not by a single number $δ^{\cal K}$, but by a whole non-decreasing sequence $\{k_j\}$, which is squeezed between two staircases, ${\rm entier}\left(\frac{j-1}{δ+1}\right)\leq k_j\leq{\rm entier}\left(\frac{j-1}{{\rm deg}\,Δ}\right)$, where the upper bound involves Alexander polynomial $Δ(q^2)$ and follows from the continuation of the Melvin–Morton–Rozansky-Kashaev limit to $A=q^{-m}$ with $m>0$, when the limit gives a \it positive power of Alexander. The old assumption $δ+1={\rm deg}\,Δ$ is oversimplified and fails for 19 non-alternating knots with 11 and 12 crossings. Last, we list the differences of non-rectangularly colored HOMFLY for various mutant pairs to demonstrate their universality (knot-independence) and nice consistency with the spirit of DE.

• 20 paź Yuhui Chen; Shunyu Wan. U torsion structure in Heegaard Floer homology. arXiv PDF

  We construct infinite families of Brieskorn spheres with three singular fibers for which the difference between the largest and the second largest length of $U$-torsion in their Heegaard Floer homology $HF^-$ increases to infinity as the family parameter increases. In particular, these families give a negative answer to the strong geography restriction problem by Alfieri and Binns \cite[Question 1.10]AB24 which was added to the recent K3 list \cite[Problem 3.55, Remark(5)]K3. As a consequence of the results by Alfieri and Binns, these families give the examples of integer homology spheres that are neither surgery on a knot nor large surgery on links.

• 19 paź Samuel Lewis-Monkman. Weak reducibility for bridge trisections. arXiv PDF

  A bridge splitting of a knot is weakly reducible if it has a pair of disjoint shadows, one from above and one from below, in the splitting surface. The process of weak reduction exchanges local minima and maxima in the splitting, untelescoping the splitting and giving a method for detecting essential surfaces in the knot complement. Bridge trisections generalize bridge splittings to the setting of surfaces in 4-manifolds. We define weak reducibility in this context, giving diagrammatic conditions for carrying out a weak reduction operation on bridge splittings. We produce a partial classification of 4-bridge trisections of surfaces in $S^4$.

• 18 paź Geunyoung Kim; Sam Nelson; Puttipong Pongtanapaisan. Biquandle Image Quivers. arXiv PDF

  We introduce a new family of weighted quiver-valued invariants of oriented classical and virtual knots and link, surface-knots and surface-links, and immersed surface-knots and surface-links associated to finite biquandles. We derive several polynomial invariants from these quivers via decategorification. We provide examples to illustrate the computation of the invariant as well as toy examples showing tables of invariant values for small biquandles.

• 18 paź Seong-Jin Lee. Negative-Definite Plumbings and New Unknotting Numbers. arXiv PDF

  We use Neumann's normal form to decide whether the double branched cover of a knot bounds a negative-definite plumbing tree in either orientation, starting from the recognised Seifert or graph-manifold structure of the cover. For many non-alternating knots with at most thirteen crossings whose unknotting numbers were unknown, this identifies the cases in which the Heegaard Floer homology of the cover can be computed from such a plumbing, and shows for most of the others that it cannot. We determine 112 new unknotting numbers. For 43 of them, the lower bound rests on Heegaard Floer homology computed from the plumbing; for the other 69, it comes from linking pairings and from a theorem of McCoy and Park on special alternating knots. Finally, we show that the knot 13n1587 is a further explicit counterexample to the Bernhard–Jablan conjecture, after 13n3370.

• 18 paź Ke Feng; Aijin Lin. Non-geometric ideal triangulations with all edge valences at least six. arXiv PDF

  Costantino, Frigerio, Martelli and Petronio conjectured that an ideal triangulation in which every edge has valence at least six can be realized by hyperbolic partially truncated tetrahedra on the compact 3-manifold it defines. We show that this is false. Starting from the figure-eight knot complement, we construct an ideal triangulation with $2688$ tetrahedra and edge valences six and eighteen whose manifold is hyperbolic with totally geodesic boundary, but which admits no such realization. The obstruction is a critical point of the covolume energy of Luo and Yang at which three tetrahedra have collapsed: convexity then excludes every geometric competitor, an Euler characteristic count excludes edges of length zero, and the critical point itself is located by a finite certificate in exact integer arithmetic. The obstruction persists in every finite cover, while a complementary theorem shows that no counterexample can have only two edge lengths or only two orbits of edges.

• 16 paź Rinat Kashaev; Vladimir Mangazeev; Marcos Marino. Analytic quantum invariants. arXiv PDF

  Using a deformed Borel-Laplace resummation, we propose a framework for repackaging quantum invariants as analytic invariants. We illustrate our approach through a detailed analysis of the coloured Jones polynomials of several simple knots, deriving explicit expressions for the resulting analytic invariants in terms of $(q,X)$-series. These invariants are closely related to, but generally distinct from, those obtained via the conventional Borel resummation of the Melvin-Morton-Rozansky expansion. As a further example, we construct an analytic invariant from a sequence of two-variable polynomials conjectured to be the Links-Gould polynomials of the figure-eight knot, associated with the quantum superalgebras $U_q(\mathfrak{gl}(n|1))$. Finally, we apply our technique to $q$-series arising in the study of three-manifold invariants, revealing connections with mock modular forms.

• 16 paź Yury Belousov; Vadim Stepaniuk. Explicit formula for the Rasmussen invariant of odd pretzel knots. arXiv PDF

  We compute the Rasmussen s-invariant for all odd pretzel knots, giving an explicit closed formula. The formula extends to all slice-torus invariants that are invariant under mutation. In the absence of mutation invariance, we prove the formula under a mild condition on the pretzel parameters, which we conjecture to be unnecessary.

• 16 paź Isidora Bailly-Hall; Michelle Chu; Karina Dovgodko; Akash Ganguly; i in. Two-bridge knot epimorphisms and intersections in the character variety. arXiv PDF

  We study the SL(2,C) character varieties of hyperbolic two-bridge knot groups admitting epimorphisms onto the trefoil knot group. Ohtsuki, Riley, and Sakuma give a systematic construction of an infinite family of such knots. The character varieties of these knot groups contain a component of irreducible representations arising from the epimorphism, which is distinct from the canonical component. We prove that every affine point of intersection between this component and any other component is algebraic non-integral, and therefore detects an essential surface in the knot complement. We also determine the boundary slope of each detected surface explicitly in terms of the continued fraction expansion of the two-bridge knot.

• 16 paź Enrique Aycart-Maldonado; Alejandro Calleja. A Topological Quantum Field Theory for Representation Varieties of Torus Links. arXiv PDF

  We construct a Topological Quantum Field Theory (TQFT) that computes the virtual classes of $G$-representation varieties associated with the fundamental group of a torus link complement. Following the framework established by González-Prieto, Logares and Muñoz for $G$-representation varieties of surface groups, we introduce a suitable bordism whose iteration enables a recursive topological evaluation of these virtual classes. Via this TQFT, we explicitly compute the virtual class of these varieties for the groups $G=\mathrm{AGL}_1(\mathbb{C})$ and $G=\mathrm{SL}_2(\mathbb{C})$.

• 15 paź Alexander Hock; Sergey Shadrin. Augmentation curve for torus knots. arXiv PDF

  We prove a version of topological recursion for the correlation differentials of colored HOMFLY-PT polynomials of torus knots on the corresponding augmentation curves, following the proposal of Gu, Jockers, Klemm, and Soroush. To this end, we explicitly construct the spectral curve from the so-called auxiliary curve of Brini, Eynard, and Mariño and prove that it coincides with the augmentation curve for every torus knot with coprime $(r,s)$. Finally, our version of topological recursion proves the higher-genus closed-string prediction of the Aganagic–Vafa conjecture for all torus knot augmentation curves.

• 15 paź Eaman Eftekhary; Marzieh Souri. Holomorphic Triangles in the Heegaard Floer Pillowcase. arXiv PDF

  The Heegaard Floer pillowcase is a recurring local model for rational tangle replacement and exact-triangle constructions. We compute the triangle product for its standard rational-slope triples of curves and encode all contributing classes in a finite generating polynomial. One slope ordering gives a vanishing map. For the other, a closed form for the generating function is computed. One may assume that the slopes are $\tfrac10,\tfrac pq$ and $\tfrac{p'}{q'}$. When an integer lies between the two finite slopes, the closed form has a particularly simple description and there are exactly $2(pq'-p'q)+2qq'$ contributing classes. We also compute the absolute Maslov bi-gradings of the intersection generators in the employed model, relating the planar labels to the graded Floer groups of two-bridge knots and links.

• 15 paź Honghuai Fang. Quantum Color Newton Rings. arXiv PDF

  We study integral Newton rings attached to symmetric quantum colors. We show that their boundary values at a deleted node are exactly the elements of Habiro's cyclotomic completion, and we describe their quantum neighborhoods at $q=1$ uniformly in the rank. The symmetric-color $\mathfrak{sl}_n$ invariants of a knot are the values of an element of such a ring. In its expansion at $q=1$, the coefficient of $(q-1)^m$ has denominator dividing the $(n-1+2m)$th power of the Alexander polynomial, with integral numerator, and this exponent is sharp. For $5_2$ and $6_1$ at $q=i$, the period matrices of the rank-two coefficient recurrences have spectral modules of exact order three in the Picard group of an affine elliptic curve.

• 15 paź Tommy Hofmann; John Nicholson. A counterexample to a conjecture of Plotnick on knot exteriors. arXiv PDF

  We resolve a conjecture of S. Plotnick from 1983 concerning the homotopy type of four-dimensional knot complements. We show that the exteriors of two branched twist spins of the trefoil, which were shown by Plotnick not to be homotopy equivalent as boundary pairs and hence not homeomorphic, are in fact simple homotopy equivalent. The main ingredient of the proof is the construction of an explicit element of augmentation $49$ in the integral group ring of $\operatorname{SL}_2(\mathbb{F}_5)$ which is a unit modulo the group norm and commutes with the element inducing the monodromy on the fundamental group.

• 14 paź Jonathan A. Higgins. Minimal $\mathfrak{sl}_N$ Complexes of Rational Tangles via the Fukaya Category. arXiv PDF

  We compute minimal $\mathfrak{sl}_N$ complexes for oriented rational tangles, denoted $[\![τ_{u/v}^{O}]\!]_N$, in terms of the topological Fukaya category of the $3$-punctured disk. This demonstrates that the complexes stabilize, allowing us to define a HOMFLY-PT homology for rational tangles.

• 14 paź Oscar Ocampo. Crystallographic quotients of Artin braid groups and torus bundles over flat $3$-manifolds. arXiv PDF

  For $n\geq3$, let $\overline B_n=B_n/[P_n,P_n]$ be the crystallographic quotient of the Artin braid group, with translation subgroup $\overline P_n=P_n/[P_n,P_n]\cong\mathbb Z^N$, where $N=\binom n2$. We prove that every compact connected flat $3$-manifold $M$, considered up to affine equivalence, is the base of a geometric torus bundle whose total space is a compact flat manifold with fundamental group a Bieberbach subgroup of some $\overline B_n$. More precisely, for every $n\geq n_M$, where $n_M\in\{3,4,5\}$ is an explicit optimal threshold depending on $M$, we construct a Bieberbach subgroup $Γ_n(M)\leq\overline B_n$ of dimension $N$ and a complete normal subgroup $K_n(M)\cong\mathbb Z^{N-3}$ consisting of translations such that $$ Γ_n(M)/K_n(M)\congπ_1(M). $$ We also determine which of the associated torus-bundle group extensions split: for the realizations constructed here, splitting occurs for exactly four of the ten affine-equivalence classes of compact flat $3$-manifolds.

• 14 paź Vyacheslav Krushkal; Ryan Stees. Relative knot concordance. arXiv PDF

  Motivated by the relative-slice problem in $4$-manifold topology, we define for a knot $R$ the notion of a relative knot concordance group $\mathcal{C}_R$ and analyze the structure of its algebraic analogue $\mathcal{AC}_R$ modulo torsion. Using Milnor signatures, we show that $\mathcal{AC}_R$ has infinite rank for every $R$. We also show that any prescribed finite set of natural numbers can be realized as orders of elements in a single relative concordance group.

• 14 paź Russell Avdek; Youlin Li; Zhengyi Zhou. Algebraic overtwistedness of contact $(+1)$-surgeries via augmented Floer theories. arXiv PDF

  We prove that contact $(+1)$-surgery along a Legendrian $n$-sphere $Λ\subset(\mathbb{S}^{2n+1},ξ_{\mathrm{std}})$ with \[ \mathrmtb(Λ)\ne (-1)^\fracn(n-1)2+1 \] has vanishing contact homology. In dimension three, this produces new examples of tight algebraically overtwisted contact manifolds, vastly extending the authors' previous results: If a Legendrian knot in $(\mathbb{S}^3,ξ_{\mathrm{std}})$ admits an exact, oriented Lagrangian filling that is not a disk, then contact $(+1/k)$-surgery on it is algebraically overtwisted and tight for every positive integer $k$. Our proofs use versions of symplectic cohomology and wrapped Floer cohomology defined by augmentations of a chain-level contact homology algebra.

• 13 paź Matthew Harper; Ben-Michael Kohli. The Links–Gould polynomial of an alternating link is alternating. arXiv PDF

  Building on work of Kauffman–Saleur and earlier work of the second author, we derive a determinant formula for the Links–Gould polynomial of links. As an application, we prove a 20-year-old conjecture of Ishii: for any alternating link, the Links–Gould invariant is an alternating polynomial.

• 13 paź Daniel Galvin; Daniel Kasprowski. Stably exotic knotted surfaces up to equivalence. arXiv PDF

  In this short note we show the existence of stably exotic knotted surfaces in 4-manifolds in the sense that the embedded surfaces are topologically equivalent but are not smoothly equivalent even after any number of internal and external stabilisations.

• 13 paź Kefeng Liu; Pan Peng; Shengmao Zhu. A proof of the framed LMOV conjecture for links. arXiv PDF

  We prove the Labastida–Mariño–Ooguri–Vafa conjecture for oriented links with arbitrary integer framings. Consequently, the framed reformulated invariants have finite integral LMOV expansions for every integer framing vector.

• 12 paź Jiayu Wan; Charles Katerba. On Intersection of Character Varieties Associated to Dehn Fillings of the Whitehead Link and Its Generalization. arXiv PDF

  The canonical component of the character variety of a hyperbolic 2-bridge link complement M is an algebraic surface, from the work of Thurston. If we fix a boundary component T of M and perform a Dehn-filling on T, we get a 1-cusped manifold whose character variety is commonly a generalized curve that is naturally embedded in the character variety of M. If we perform distinct Dehn-fillings on M, we get various generalized curves on the canonical surface. In this paper, we analyze how these curves intersect with each other in the surface and study the topological implications for these intersection points. We also classify the intersection points that correspond to reducible characters, using the twisted Alexander polynomials associated to the Dehn-fillings. Throughout the paper, we use the Whitehead link as a motivating example to illustrate our core methodology, while some generalizations to other 2-bridge links will also be discussed.

• 12 paź Mehmetcik Pamuk. Homotopy Self-Equivalences of $\mathbb{R}P^2\times S^2$ and a Braid for non-orientable $4$-Manifolds. arXiv PDF

  We compute the group of based homotopy classes of self-homotopy equivalences of the non-orientable $4$-manifold $\RP^2\times S^2$. It is an elementary abelian $2$-group of rank $5$. The subgroup of classes whose based lift to the universal cover $S^2\times S^2$ has degree $+1$ has rank $4$ and is isomorphic to the group of free homotopy classes. The group maps onto the group of self-equivalences of the algebraic $2$-type, which has order $8$, with kernel $π_4(\RP^2\times S^2)\cong(\Z/2)^2$ generated by pinch-fold maps; that these maps are independent is shown by an adjunction argument. We then construct a non-orientable analogue of the Hambleton–Kreck braid of exact sequences in dimension $4$, for non-orientable manifolds with spin universal cover, stated for finite fundamental group. We apply the braid to $\RP^2\times S^2$ and to $\RP^4$, and for the Hambleton–Hillman manifold with fundamental group $(\Z/2)^2$ we determine the normal $1$-type and show that self-equivalences act on $π_2$ through diagonal sign matrices. For $\RP^2\times S^2$ the pinch-fold maps are detected by twisted spin bordism, and, assuming facts from surgery theory stated as hypotheses, exactly one of the three non-trivial pinch-fold maps is homotopic to a homeomorphism.

• 12 paź Werner Stanggassinger. The symmetry group of Turk's head knots. arXiv PDF

  For p, q >= 3 with gcd(p,q) = 1 we show that the symmetry group of the Turk's head knot Th(p,q), the closure of the braid (sigma_1 sigma_2^-1 sigma_3 sigma_4^-1 ... sigma_p-1^+-1)^q, is the dihedral group of order 4q. This proves Conjecture R of Di Prisa and Şavk for q >= 3; the case p = 3 is due to Sakuma and Weeks. The argument is diagrammatic: by the flyping theorem of Menasco and Thistlethwaite, the symmetries of a knot with a prime, reduced alternating diagram without non-trivial flypes are the symmetries of the diagram (we write out a proof for the diagrams at hand), and we show that the standard diagram of Th(p,q) has no non-trivial flypes and exactly 4q symmetries. The chirality of Th(p,q) for even p, which also follows from a theorem of Murasugi, is visible in the group. We compare the argument with the geometric one of Sakuma and Weeks for p = 3.

• 12 paź Zhechi Cheng. Knot Floer homology of fibered positive links. arXiv PDF

  We determine the Maslov-graded next-to-top term of knot Floer homology for fibered positive links. Its rank in Maslov grading (-1) is the number of nontrivial prime connected-sum factors plus the number of components minus one, and the remaining summand is supported in Maslov grading zero. For a knot, the dimension of this remaining summand is determined by its Alexander polynomial. The proof combines a preferred-crossing construction with reductions of generalized Seifert surfaces to planar pretzel surfaces and a graded skein induction. We also obtain bounds on the fixed points of monodromies for fibered positive knots.

• 12 paź Jun Wang; Shasha Yang. An Algebraic Matching Algorithm for the Genus of Grid Diagrams. arXiv PDF

  Grid diagrams encode knots and links combinatorially through an (n,n) winding matrix W(G). A fundamental invariant of the grid diagram is its genus g(G), which records essential topological information about the underlying knot. We give an algebraic algorithm that computes g(G) directly from W(G). The method repurposes the Hungarian algorithm ,after a column normalization that makes the matrix non-negative, it applies row-column reductions and minimum-vertex-cover corrections until the zero entries admit a perfect matching, equivalently until the minimum marking number equals n. By Konig's theorem the resulting permutation defines a distinguished grid state, and g(G) is then read from this matrix via the classical grid homology formula (Ozsvath-Stipsicz-Szabo). The novelty of this paper is the algorithm and its strongly polynomial analysis; the underlying invariant is known grid-homology theory. We illustrate the method on the trefoil and additional examples, and discuss extensions to (m,n) matrices and grid-move navigation toward the knot genus g(K).

• 12 paź Seiichi Kamada. Loop Fundamental Quandles: A Geometric Generalization of the Knot Quandle. arXiv PDF

  The knot quandle, introduced independently by Joyce and Matveev, admits a geometric description in terms of nooses. Motivated by this description, we introduce the \textitloop fundamental quandle $LQ(X,A,\ast)$ of a pair $(X,A)$ with a fixed base point $\ast\in X$. Its elements are represented by pairs consisting of a loop in $A$ and a path joining the base point of the loop to the fixed base point $\ast$. More generally, for a loop condition $P$ on $A$, invariant under free homotopy, we define the loop fundamental quandle $LQ(X,A,\ast;P)$. We establish functoriality and homotopy invariance of the loop fundamental quandle and show that it admits a natural augmented quandle structure over $π_1(X,\ast)$. For a knot $K$ in $S^3$, applying the construction to its exterior gives the \textitknot loop quandle $LQ(K,\ast;P)$. Appropriate choices of the loop condition recover the knot quandle, its $n$-th power quandles, and the full knot quandle. The construction is inspired by Joyce's noose-theoretic approach, but allows loop data that need not arise as the boundary of a disk satisfying the noose condition.

• 12 paź Yuki Matsushima. On infinite knot families with the same HOMFLYPT polynomial. arXiv PDF

  In this paper, we focus on the examples constructed by Kanenobu and provide new family knots with the same HOMFLYPT polynomial invariants, which are distinguished by the Kauffman polynomial.

• 10 paź Weizhe Shen. Shake concordance in the topological concordance class of the Hopf link. arXiv PDF

  Within the topological concordance class of the Hopf link, we construct infinitely many classes under the equivalence relation generated by smooth $0$-shake concordance. Each contains infinitely many smooth concordance classes represented by links that are pairwise strongly $0$-shake concordant and whose second components have unbounded smooth slice genus.

• 9 paź John B. Etnyre; Hyunki Min; Bülent Tosun; Konstantinos Varvarezos. Tight surgeries on torus knots. arXiv PDF

  We give a complete classification of tight contact structures on manifolds obtained from any surgery along any torus knot, which provides the first such classification on an infinite family of knots. The main tools are the classification of non-loose torus knots, bordered contact invariants via their immersed curve interpretations, and non-thickenable tori in knot complements. As a byproduct of this classification, we also give the first classification of contact structures on some surface bundles over the circle with fibers of genus larger than $1$. The symplectic fillability of these contact structures is also studied and leads to, among other things, the disproof of a conjecture of Ghiggini and Van Horn-Morris on Stein fillability of some contact structures on a Brieskorn sphere.

• 9 paź Indranil Biswas; Shrinit Singh. Recognizing oriented knots from finite quotients with one marked element. arXiv PDF

  For an oriented tame knot $K$, let $G_K$ be its group, with positive meridian $μ_K$ and preferred longitude $λ_K$. We prove that the profinite completion $\widehat{G_K}$, marked by the conjugacy class of $μ_Kλ_K$, determines $K$ up to oriented equivalence. In fact, every isomorphism preserving this mark preserves the meridian and longitude with the same conjugator. The proof combines Xu's genuineness theorem with profinite JSJ theory and simultaneous boundary realization, and applies to satellite and composite knots. Consequently, Eisermann's colouring polynomials over all finite groups determine oriented knots.

• 8 paź Jacob Guynee. Quasipositive Braids, Presentations, and Garside Structures. arXiv PDF

  We provide a presentation of the braid group in which the generating set is the set of all positive half-twists, the relations are balanced positive relations, and every relation is conjugate to one in a finite list. We then prove that for any such presentation, its positive monoid is not isomorphic to the quasipositive braid monoid. Finally, we show that the quasipositive braid monoid has no proper Garside family. These results settle a problem of Calegari from the K3 problems list.

• 8 paź Matthew Harper; Ben-Michael Kohli. The Colored Jones Function does not determine any colored Links–Gould polynomial. arXiv PDF

  We prove that there is no uniform description of $\mathrm{LG}^{(n)}$ in terms of colored Jones polynomials for all knots, even when allowing infinitely many colors and arbitrary operations. We prove the $n=1$ case, using an existing cabling formula and the separation of the Conway and Kinoshita–Terasaka knots by $\mathrm{LG}^{(2)}$. We develop an odd cabling formula and use a Vandermonde argument to show that for each $n\geq 2$ some satellite of each of these knots is distinguished by $\mathrm{LG}^{(n)}$.

• 8 paź Boning Wang. A note on realizing triples of slice, concordance and Seifert genera. arXiv PDF

  We show that every triple of integers $(a,b,c)$ satisfying $1\leq a\leq b\leq c$ is realized as the smooth slice genus, smooth concordance genus, and Seifert genus of a knot, respectively. This proves a conjecture of Kearney.

• 8 paź Ian Hambleton; Kursat Sozer. Braidings of Self-Equivalences and Bordism. arXiv PDF

  Let $M$ be a closed, smooth or topological $n$-manifold, with $n \geq 4$. We construct a homotopy highly cartesian square relating the space ${\mathcal E}(M (\ell))$ of homotopy self-equivalences of $M$ (in a suitable range) over the Postnikov $\ell$-sections of its stable normal microbundle, and an $(\infty + n)$-fold loop space representing an associated (normal) bordism theory. This implies the existence of braids of interlocking exact sequences involving the homotopy groups of ${\mathcal E}(M(\ell))$ and certain Lashof bordism groups, leading to a conceptual explanation and broad generalization of earlier work of Hambleton–Kreck for closed, oriented $4$-manifolds.

• 7 paź Gabriel Corrigan; Isacco Nonino. Motion groups, Legendrian links, and non-trivial loops of embeddings. arXiv PDF

  We introduce, for an embedded Legendrian link $L$ in a contact 3-manifold $(M,ξ)$, the group of its Legendrian motions. Analogously with the smooth case, we prove that this Legendrian motion group is isomorphic to the fundamental group of the space of unparametrised Legendrian embeddings. For all Legendrian knots in the standard contact 3-sphere, we characterise Legendrian motions in terms of contact mapping classes of the knot exterior. In an appendix, we give a short exposition of our techniques applied to the smooth case, comparing to the classical Dahm homomorphism.

• 7 paź Valentina Bais; Porter Morgan; Daniele Zuddas. Lifting braids over $\mathbb{RP}^2$ and non-orientable Lefschetz fibrations. arXiv PDF

  We show that, given a closed connected non-orientable surface $F$ and a simple branched cover $p \colon F \rightarrow \mathbb{RP}^2$ of degree $d \geq 5$, every self-diffeomorphism of $F$ is isotopic to the lift of a self-diffeomorphism of $\mathbb{RP}^2$ with respect to $p$. As an application, we show that every Lefschetz fibration over a connected surface $B$ with $\partial B\neq \emptyset$ and closed non-orientable regular fiber can be expressed as the composition of a simple cover of $B\times \mathbb{RP}^2$ branched over a surface braided over $B$ and projection onto the $B$-factor.

• 7 paź Viktória Földvári; Vera Vértesi. Bounds on Legendrian Double Twist Knots. arXiv PDF

  We study Legendrian realizations of double twist knots in the standard tight contact 3-sphere. For the family K(4,m), we give upper bounds on the number of oriented and unoriented Legendrian isotopy classes of maximal Thurston–Bennequin representatives. The proof uses bypass techniques in convex surface theory and the classification of tight contact structures.

• 7 paź Alfredo Guevara. Gluons and Knots. arXiv PDF

  We argue that the scattering amplitude of a single minus helicity gluon with arbitrarily many positive helicity gluons is topological. The amplitude is constant in chambers of kinematic space, its resonances are located at walls and its factorization is determined by wall-crossing. Yangian invariance dictates the amplitude to be determined by a 2d null Wilson loop that can be lifted into a positive and transverse 3d link. By reinterpreting wall-crossing and the Weinberg soft theorem as skein relations and Reidemeister moves of the subknots, we construct a state-counting description consisting of adding the `corners' of their HOMFLY-PT polynomials. Beyond the classical limit, we argue that this provides a conjectured but precise connection to topological field and string theory. The new formula is rigorously derived via a Lean-formalized dependency graph (DAG) as detailed in a companion paper, laying out the precise assumptions including Reidemeister III invariance. It is further tested exhaustively, including novel `star' kinematics where the amplitude and state count are shown to become Catalan numbers, in agreement with knot theory literature.

• 7 paź Pablo Portilla Cuadrado. A reduction theorem for Lê's conjecture. arXiv PDF

  Let $\boldsymbol{n}=(n_1,n_2,n_3):(\mathbb{C}^2,0)\to(\mathbb{C}^3,0)$ be a holomorphic map germ admitting an injective representative. We prove that if $(d\boldsymbol{n})_0=0$, then $\mathrm{ord}_0(\boldsymbol{n})\in\{2,3,4\}$. This reduces Lê's conjecture to excluding potential counterexamples $\boldsymbol{n}$ of orders $2$, $3$, and $4$. The proof combines techniques from the theory of plane curve singularities, explicit cobordism constructions, and genus bounds obtained by applying properties of the $Υ$ invariant coming from knot Floer homology.

• 7 paź Alice Merz; Diego Santoro. Zero-surgery characterises Whitehead doubles. arXiv PDF

  We show that zero is a characterising slope for the untwisted Whitehead double of any knot $K$. More generally, this holds for all $n$-twisted Whitehead doubles of $K$, with the assumption that $n\ne pq$ if $K$ is a non-trivial $(p,q)$-torus knot or $(p,q)$-cable. In particular, zero is a characterising slope for all twist knots.

• 7 paź Yutong Dai; Oliver Hayman; András Juhász; Ludovico Morellato. Computations of the slice genus and the unknotting number of links via machine learning. arXiv PDF

  Links are disjoint unions of circles smoothly embedded in $S^3$. We use reinforcement learning and Bayesian optimisation to obtain new upper bounds on several link invariants that are not known to be algorithmically computable: the slice genus and the unknotting number for links, and the strong slice genus for algebraically split links. We also compute lower bounds using known invariants. Combining the upper and lower bounds, we obtain new exact values in many cases. Our unknotting agents can reproduce the non-additivity of the unknotting number for several counterexamples due to Brittenham and Hermiller, in some cases finding new unknotting trajectories.

• 7 paź Kazuhiro Ichihara; Kengo Kishimoto. Purely cosmetic surgeries on knots with low-span Jones polynomials, II. arXiv PDF

  We show that a knot in the 3-sphere with a nontrivial Jones polynomial of span at most 14 admits no purely cosmetic surgery. This extends a result of the first author for span at most 11. We determine all Laurent polynomials of span 15 satisfying the conditions on the Jones polynomial used in the proof. These polynomials form an infinite family. We also show that a knot of braid index at most 4 whose Jones polynomial has odd span admits no purely cosmetic surgery.

• 7 paź Cristina Anghel; Daniel López Neumann; Roland van der Veen. The top coefficient of the Links-Gould invariant is Murasugi-sum multiplicative. arXiv PDF

  We prove that the Laurent polynomial in $\mathbb{Z}[q^{\pm 1}]$ that is the top coefficient of the Links-Gould invariant of the boundary of a Seifert surface is multiplicative under arbitrary Murasugi sums of minimal genus Seifert surfaces. This extends a previous result of the second and third authors which only holds for plumbing, i.e. 4-Murasugi sum. Our technique relies on quantum supergroup themselves and PBW basis rather than the skein relations used in the previous work.

• 7 paź Yangxiao Luo; Shunyu Wan. On unifying Jones representations of braid groups and Thompson's group. arXiv PDF

  We unify Jones' representations of braid groups and the oriented Thompson's group by introducing the oriented dyadic braid group. This finitely generated group contains every braid group and the oriented Thompson's group. We construct an embedding of the oriented Thompson's group into this group such that taking braid closures recovers Jones' construction of oriented links. We also introduce the dyadic Temperley-Lieb algebra and construct a unitary representation of the oriented dyadic braid group extending the classical Jones representations of braid groups. Along the Thompson group's embedding, its distinguished matrix coefficient agrees with the corresponding coefficient of Jones' representation of the oriented Thompson's group.

• 6 paź Chen Zhang. Cabling formula and colored knot Floer homology. arXiv PDF

  We study colored knot Floer homology via a bordered cabling construction. We construct a cabling module and identify the connecting maps in the defining direct system at the level of this module. By taking the colimit at the chain level, we obtain a general formula for colored knot Floer homology of an arbitrary knot in terms of its ordinary knot Floer homology and the colored unknot algebra. This extends the computation of Alishahi-Gorsky-Liu from L-space knots to all knots.

• 6 paź Naageswaran Manikandan. Reduced even and odd Khovanov homology of positive and $0$-adequate links. arXiv PDF

  In this article, we study reduced even and odd Khovanov homology, with a particular focus on positive and $0$-adequate links. For positive links, we show that the first reduced homology is supported in a single quantum grading, where it is free abelian, and that its rank is determined by the Seifert graph of any positive diagram. As a consequence, reduced even and odd Khovanov homology, as well as unreduced odd Khovanov homology, detect fiberedness among positive links, extending the previously known result for unreduced even Khovanov homology. We further show that the torsion in these three theories does not detect fiberedness in the same way as the torsion in unreduced even Khovanov homology. We further show that $(p,q)$-cables of positive knots with $q\geq p$ exhibit the same behavior, even though such cables need not themselves be positive. We also study the stable homotopy types associated to reduced even and odd Khovanov homologies in the extremal and almost extremal quantum gradings of $0$-adequate links. At the almost extremal quantum grading, we show that the reduced stable homotopy types are wedges of sphere spectra. This contrasts with the unreduced setting, where the corresponding homotopy types depend on whether the $0$-state graph is bipartite.

• 6 paź Ya. Kononov; A. Morozov. What a knot sees at a root of unity. arXiv PDF

  HOMFLY-PT polynomials describe analytic continuation of Wilson averages in Chern-Simons theory to arbitrary complex values of two independent parameters $q$ and $A$. They also depend on the knot and representation of the gauge group. However, when $q^2$ is a primitive $m$-th root of unity, e.g. $q=\pm e^{iπ/m}$, these polynomials exhibit a universal (knot-independent) factorization: $H_R$ becomes the product of the polynomials for the $m$-core of the Young diagram $R$ and of one and the same factor $H_{[m]}$ for every $m$-ribbon of $R$. In all the examples which we computed this factor is just the special polynomial at $A^m$. At $q=\pm1$ factorization follows from the singularity of the cabled HOMFLY, which is the same for all representations. At $m>1$ the relevant singularity is in the projector, and what survives of it is a finite sum of braids $X_m$ – the $m$-th power sum inside the cable, which can be freely moved through the other strands. For the proof we use the Murnaghan-Nakayama rule and the Adams operation. Discovery of this hidden structure explains the old factorization puzzle.

• 5 paź Stavros Garoufalidis; Shana Yunsheng Li. The colored Jones polynomial does not determine the Links–Gould polynomial. arXiv PDF

  We prove that the colored Jones polynomial of a knot does not determine the Links–Gould polynomial (associated with the 4-dimensional representation of the Lie superalgebra $\mathfrak{sl}(2|1)$) with two different approaches: the first is to find a concrete genus 2 mutation pair with different Links–Gould polynomials, and the second follows from the fact that the colored Jones polynomial is invariant under Conway mutation whereas the colored Links–Gould polynomial is not. The first approach also applies to a related family of polynomial invariants of Kashaev–Mangazeev, which are conjectured to be the $\mathfrak{sl}(n|1)$-knot invariants colored by the $2^n$-dimensional irreducible representation.

• 5 paź Hamid Abchir; Khaled Qazaqzeh; Mohammed Sabak. On Alternateness of the Jones-Kauffman Polynomial of Virtual Links. arXiv PDF

  We study alternateness of the Jones-Kauffman polynomial of virtual links. In particular, we identify a class of virtual links with alternating Jones-Kauffman polynomial that will be called the class of mock-alternating virtual links. Then, we explain how every mock-alternating virtual link can give rise to an infinite family of mock-alternating virtual links using the twisting technique introduced in [4] and later generalized in [5,18]. Finally, we give a connected state expansion of the mock-determinant for checkerboard-colorable links.

• 3 paź Stavroula Makri. Mixed configuration spaces, fixed points and Loop braid groups. arXiv PDF

  We introduce the notion of mixed configuration spaces of the $3$-ball. That is, we consider both distinct circles and distinct points in the interior of the $3$-ball and we prove that the projection onto the configuration space of circles of the $3$-ball is a locally trivial fibration. The key idea for defining the mixed configuration spaces is to study fixed points of orientation-preserving homeomorphism of the $3$-ball that leave invariant a trivial link of $n$ components in the interior of the $3$-ball. In particular, we prove that two fixed points are Nielsen equivalent if and only if the associated loop braids are conjugate by an element of a distinguished free subgroup of rank $n$. This result stands as a $3$-dimensional counterpart of the $2$-dimensional result where fixed points of homeomorphisms of the punctured disc are characterized in terms of braid elements of the classical Artin braid group.

• 2 paź Sai Kambampati. Integral face-edge duality for knot triangulations. arXiv PDF

  For an ordered ideal triangulation of a knot exterior in $S^3$, we give a cochain construction relating its face matrix $C$ to its preferred-longitude gluing matrices $A,B$. The obstruction to uniqueness is the first cohomology of the ideal compactification. Its vanishing yields nullity doubling and the FAMED matrix identity, including a constrained formulation for singular face matrices. A separate rank argument under a strict angle structure supplies a proposed proof of Wong's generalized FAMED conjecture. Over the integers, the construction gives a direct-sum decomposition of $\ker_{\mathbb{Z}}B$ and an exact sequence from the two face cokernels to the gluing cokernel, with quotient $\mathbb{Z}/2\mathbb{Z}$. In the unhalved longitude convention this implies $|\det B|=2|\det C|^2$. We also obtain cyclic face torsion, order-reversal invariance of Smith normal form, and affine face coordinates for fixed lifted gluing systems. The determinant-one assertion and the full Andersen–Kashaev volume conjecture remain open here.

• 1 paź James Hughes; Jiajie Ma. Cluster structures from Legendrian double twist knots: a comparative approach. arXiv PDF

  We undertake a systematic comparison of cluster algebras associated to Legendrian knots of the same smooth knot type. We describe a cluster structure on the m-graded augmentation variety of Legendrian representatives of a family of double twist knots that includes all orientably Lagrangian fillable twist knots. For different Legendrian representatives of the same smooth knot type, we show that these cluster structures are equivalent. Along the way, we characterize the representatives that admit Maslov-m exact Lagrangian fillings, construct a Catalan number of such fillings for each representative, and show that any Maslov-0 fillable representative is Legendrian isotopic to the (-1)-framed closure of a positive braid. Finally, we give two Legendrian representatives of a cabled twist knot that yield equivalent cluster structures of infinite type.

• 1 paź Alexei Vernitski. Some petal diagrams of the unknot are hard. arXiv PDF

  A petal diagram of a knot is a projection with a single multi-crossing and no nested loops; it is encoded by a permutation of the heights of the strands through the multi-crossing. Colton, Glover, Hughes and Sandberg proved a Reidemeister-type theorem for petal diagrams: two petal permutations represent the same knot if and only if they are related by trivial petal additions and deletions and by crossing exchanges. We ask whether every petal diagram of the unknot can be reduced to the one-petal diagram without ever increasing the number of petals, in analogy with Dynnikov's monotonic simplification theorem for rectangular diagrams. By an exhaustive, certified computer search we show that this is true for diagrams with at most 7 petals and false for 9 petals. Of the 40320 petal diagrams with 9 petals, 24992 represent the unknot, and exactly 108 of them are hard: none of them admits a crossing exchange or a trivial petal deletion, even if two natural petal-number-preserving symmetries are allowed. Up to these symmetries and mirror image there are three hard diagrams. Two of them can be untangled by passing through 11 petals; the third cannot be untangled through diagrams with at most 11 petals, but can through 13. We explain why the phenomenon differs from the rectangular case: a petal diagram is an arc presentation whose cyclic order of pages is determined by the order of its vertices on the binding, and no elementary move of Cromwell and Dynnikov preserves this rigid structure.

• 30 wrz Anar Akhmedov. Braided Surfaces and Small Exotic 4-Manifolds. arXiv PDF

  Let $K$ be the trefoil or figure-eight knot and let $M_K$ denote zero surgery on $K$. We prove that the fiberwise elliptic involution gives a two-fold branched cover \[ M_K\times S^1\longrightarrow T^2\times S^2 \] whose branch locus is a section together with a connected braided three-section. The corresponding braid words are computed explicitly for both knots and make visible the relation with Luttinger surgery along product Lagrangian tori. More generally, for a genus-$g$ fibered knot with hyperelliptic monodromy, the same construction gives a braided $(2g+2)$-section; this includes every genus-two fibered knot. We also state the corresponding surface-bundle version over $T^2$. We then track the induced peripheral data and gluing diffeomorphisms in the constructions of $Y_K$ and its twisted double $X_K$, as studied by the author in \cite[Section 4]Akh07, and through the constructions of the small exotic manifolds homeomorphic to $\CP^2\#k\CPb^2$, $k=3,5$ \citeAkhmedovSmall,ABP,AP2008. In the symmetric construction of a $k=5$ example \cite[Section 3.3]ABP, each summand admits a double cover branched along a surface, although the marked complements used in the twisted double need not be invariant. Finally, we prove boundary-action, relative-homology, and local-smoothing obstructions to extending these local involutions across the gluing regions to a branched cover of the closed manifold.

• 30 wrz Kazumichi Nakamura. Montesinos Knots with Delta-Unknotting Number One. arXiv PDF

  The $Δ$-unknotting number for a knot is defined as the minimum number of $Δ$-moves needed to deform the knot into the trivial knot. In this paper, we discuss Montesinos knots whose $Δ$-unknotting number is equal to one. We propose a conjectural characterization of Montesinos knots with $Δ$-unknotting number one and provide examples admitting distinct $Δ$-moves, each of which deforms the knot into the trivial knot.

• 30 wrz Misha Schmalian; Steven Sivek. Lens space surgeries on knots of small genus. arXiv PDF

  We prove that the $(-2,3,7)$ pretzel knot is the only hyperbolic knot of genus at most $5$ with a lens space surgery. Our key technical result asserts that if a genus-$g$ hyperbolic knot $K$ has an elliptic surgery of slope greater than $4g-3$, then the invariant foliations of its monodromy have an orientation that is reversed by the monodromy, and so the opposite of the dilatation must be a root of the Alexander polynomial of $K$. Combined with prior work of Moser, Wu, Bleiler–Litherland, Baker, and Greene, this verifies the Berge conjecture for knots of genus at most $5$.

• 29 wrz Masahico Saito; Emanuele Zappala. Surface Braid Invariants from Yang-Baxter Operators and Their Cohomology. arXiv PDF

  Surface braids are a generalization of braids to surfaces in the 4-disk. They are described by planar graphs called charts, and sequences of braid words called braid systems, that represent monodromies of branch points in projections. Yang-Baxter operators (YBOs) have been used extensively in knot theory for constructions of Jones and other polynomials, defined by braid group representations. We propose to use YBOs for constructing surface braid invariants through braid systems. The intersections of eigenspaces are used for this construction. We further use 3-cocycles of the Yang-Baxter cohomology theories via charts, in an analogue of quandle cocycle invariants but more generally utilizing tensor products of modules. An approach towards defining knotted surface invariants through closures of surface braids is discussed.

• 29 wrz Taehee Kim; Takayuki Morifuji. Quantitative finiteness of monic characters of knots. arXiv PDF

  Dunfield, Friedl, and Jackson showed that if the $SL(2, \mathbb{C})$-character variety of a knot has an irreducible curve component that contains the character of an irreducible representation and a character with nonmonic twisted Alexander polynomial, then this component has only finitely many characters with monic twisted Alexander polynomials. In this paper, we give explicit upper bounds on the number of such characters in terms of a presentation of the knot group. In particular, we give upper bounds in terms of the crossing number of the knot.

• 28 wrz Soheil Azarpendar; Colin McCulloch. Higher Alexander Polynomials of Satellite Links. arXiv PDF

  We present formulas for the higher Alexander polynomials of satellite links.

• 28 wrz Sofia Lambropoulou; Lucrezia Beatrice Lorenzi. On pure braids and bonded braids. arXiv PDF

  In this paper we investigate the relation between the bonded braid monoid BBn and the group of pure braids $P_n$, realised by an algebraic analogue of the topological tangle insertion, namely by replacing a bond by (a power of) a pure braid generator. To do so, we provide appropriate presentations for $P_n$ using cyclic relations and clasp generators, and extend them to presentations of $B_n$. We construct the monoid epimorphism $Φ: BB_n \to Bn$, mapping bonds to pure braid generators, which is the main result of the paper. Using the isomorphism from the singular braid monoid $SB_n$ to the tight bonded braid monoid $BB_n$, we extend $Φ$ to a new type of epimorphism $s Φ: SB_n \to B_n$. The epimorphisms $Φ$ and $s Φ$ are extended to families of epimorphisms $Φ_p$, $s Φ_p$, $p \in \mathbb{Z}$.

• 28 wrz Eiichiro Hakamada. On intersection forms of spin $4$-manifolds with AR homology sphere boundary. arXiv PDF

  In this paper, building on the work of Dai–Sasahira–Stoffregen, we derive constraints on the intersection forms of smooth, compact, indefinite, spin $4$-manifolds bounded by AR homology spheres using two methods. First, we use the constraints on the existence of $\operatorname{Pin}(2)$-equivariant maps between representation spheres, established by Hopkins–Lin–Shi–Xu in their proof of the $10/8 + 4$-inequality. Second, we compute Lin's $κo$-invariant for AR homology spheres. Combined with Lin's version of the relative $10/8$-inequality, this computation yields constraints different from those based on Hopkins–Lin–Shi–Xu. As applications, we obtain new lower bounds on the minimal genus of surfaces bounded by knots whose double branched covers are AR homology spheres in various $4$-manifolds, and provide an obstruction to smooth $4$-manifold decompositions related to the $11/8$-conjecture.

• 28 wrz Nicolas Grunder. Braid Stability and a Floer theory of Braid Isotopies. arXiv PDF

  We build a Floer theory on the space of braids of periodic orbits of a Hamiltonian flow on a closed symplectic surface. The differential and continuation maps are defined by counting Floer isotopies of braids: tuples of Floer cylinders with pairwise disjoint graphs. With this new perspective, we prove a quantitative braid stability result that shows the persistence of braids under possibly large Hamiltonian perturbations. As an application, we show that for every $α\geq 0$ there is a sequence of Hamiltonian diffeomorphisms $φ_k$ on the two-torus $T^2$ such that $$d_H(\operatornameEnt_\leq α(T^2,ω),φ_k)\to \infty \quad (k \to \infty),$$ where $\text{Ent}_{\leq α}(T^2,ω)\subset \text{Ham}(T^2,ω)$ denotes the set of Hamiltonian diffeomorphisms with topological entropy at most $α$ and $d_H$ the Hofer metric. We prove this result by studying only contractible periodic orbits, whereas analogous higher-genus statements were previously obtained using non-contractible orbits.

• 28 wrz Honghuai Fang; Tian Zhou. Rectangular Cyclotomic Expansions of Colored HOMFLY-PT Polynomials. arXiv PDF

  We prove the rectangular cyclotomic expansion conjecture of Kameyama, Nawata, Tao, and Zhang for reduced colored HOMFLY-PT polynomials of arbitrary zero-framed knots. We also prove their explicit double-twist formula for every rectangular color and all pairs of nonzero integer twist parameters. For each knot, we construct a double Newton expansion with coefficients in $\mathbb Z[A^{\pm1},q^{\pm1}]$. These coefficients are uniquely determined and independent of the width and height of the rectangular color. The construction uses finite-rank representation symmetries and integral interpolation. The double-twist formula follows from a simultaneous exterior-power comparison of the two four-point Casimir operators and the known one-row formula. Finite transition formulas express the double Newton coefficients of double-twist knots in terms of the prescribed twist coefficients.

• 28 wrz Suman Saurabh. Exceptional pretzel knots are not algebraically slice. arXiv PDF

  We consider the three-strand pretzel knot $P_a=P(a,-a-2,-(a+1)^2/2)$ and prove that, when $a\equiv1\pmod4$, its Alexander polynomial fails the Fox–Milnor condition. The same remains true after adding any number of opposite odd parameter pairs $q,-q$ and reordering the parameters. Combined with work of Kim–Lee–Song, this gives the same conclusion for every odd $a\ge3$. In particular, no knot in Lecuona's exceptional family is algebraically or topologically slice. Together with earlier work of Lecuona, Miller, and Kim–Lee–Song, this completes the slice–ribbon and topological-sliceness classifications for three-strand pretzel knots and removes the nonexceptional hypothesis from the Lecuona–Wand classification of prime fibered ribbon pretzel knots.

• 27 wrz Sergey A. Melikhov. Satellites and invariants of links. arXiv PDF

  An invariant $v$ of $m$-component links is called "cableable" if there exists a $k$ such that whenever a link $L'$ is obtained from a link $L=(K_1,\dots,K_m)$ by replacing each knot $K_i$ with its $(p_i,q_i)$-cable for some $p_i$ and $q_i$, we have $v(L')=(p_1\cdots p_m)^kv(L)$. The following problem is implicit in a number of papers by P. M. Akhmetiev and originates from the Arnold-Moffatt program for finding topological lower bounds for the energy of a magnetic field: Does there exist a cableable finite type invariant of links in $S^3$ which is not a function of the pairwise linking numbers? According to Akhmetiev, he solved this problem himself, with the desired invariant being defined for $3$-component links as an analytic expression involving a magnetic field modeled on the given link. We offer a different solution, with the desired invariant for $m$-component links, $m\ge 3$, being a function of the coefficients of the Conway polynomial of the link and its sublinks; the function is given explicitly when $m=3$. Moreover, we show that the cables can be replaced by arbitrary satellites. Much of the proof is a study of low degree coefficients of the Conway potential function $Ω_L(x_1,\dots,x_n)$ expanded as a formal power series in Conway's variables $z_i=x_i-x_i^{-1}$. We also discuss type $n$ invariants which are "cableable up to an invariant of type $n-1$", some cableable invariants which are not of finite type (particularly a certain modification of Milnor's $\barμ$-invariants), and applications to links of solenoids.

• 26 wrz Anar Akhmedov; Azer Akhmedov. Higher Braided Mixed Sums, Intersection Lattices, and Symplectic Four-Manifolds. arXiv PDF

  We study mixed symplectic sums along connected unbranched cyclic multisections of degrees $p,q\geq 2$ in ruled surfaces. With \[ p(g-1)=q(h-1)=n, \] the resulting four-manifold has signature zero and Euler characteristic \[ e=4n\left(1-\frac1p-\frac1q\right). \] We classify the types having the rational cohomology ring of $\#_k(S^2\times S^2)$. They occur only for odd $k$ and, with $t=(k+1)/2$, $a=g-1$, and $b=h-1$, are exactly the solutions of \[ n=a+b+t,\qquad a\mid b+t,\qquad b\mid a+t. \] For each odd $k$ there are finitely many types; we give an explicit parametrization and tables for $k\leq 15$. Every type admits an equatorially product-framed symplectic realization and, possibly with different framings, a spin realization whose free integral intersection form is $kH$. We prove minimality, boundary incompressibility, an integral Mayer–Vietoris presentation, and a regular cyclic cover of degree $\operatorname{lcm}(p,q)$. The boundary subgroup $π_1(Σ_G)\times\mathbb Z$ embeds in the fundamental group; this gives a nontrivial Bass–Serre splitting and rules out word-hyperbolicity, cocompact real- or complex-hyperbolic lattices, and torsion-free cocompact irreducible bidisk lattices. There are five types for $k=3$ and seven for $k=5$; for $k=1$ we recover the degree pairs $(2,3)$, $(2,4)$, and $(3,3)$.

• 26 wrz Kazuhiro Ichihara. Purely cosmetic surgeries on knots with low-span Jones polynomials. arXiv PDF

  In this paper, we show that a knot with a nontrivial Jones polynomial of span at most 11 does not admit purely cosmetic surgery.

• 26 wrz Qilong Guo. Lens Space Surgeries and the Bleiler-Litherland Conjecture. arXiv PDF

  We prove the Bleiler-Litherland conjecture: every lens space obtained by a nontrivial Dehn surgery on a hyperbolic knot in $S^3$ has order at least 18. The key ingredient is a spectral obstruction: if a hyperbolic knot $K$ of genus $g$ admits a lens space surgery with slope $\pm(4g-2)$, then $Δ_K(t)$ has a real root outside the unit circle. The proof combines Floer-theoretic restrictions on lens space surgeries, Gabai's degeneracy-slope bound and Gabai-Oertel's persistence theorem for essential laminations, Ni's fixed-point theorem for monodromy, and a mod-2 orientability criterion, together with the relation between homological monodromy and the Alexander polynomial. As a further application of this spectral obstruction, we obtain characterizing-slope results for torus knots.

• 25 wrz Hakan Akgün; Xianquan Yan; Kehan Liu; Zhaoyun Chen; i in. KnottedGraph: Scalable knotted-graph topology for scientific and mathematical discovery. arXiv PDF

  Scientific data span heterogeneous structures, including coordinates, networks, surfaces, volumes and fields, yet their topology can be quantified within a common framework through graph connectivity, cycle structure, genus and spatial embedding. Graph- and homology-based summaries do not determine spatial embedding, while standard knot and link polynomials require extensions to accommodate branching graphs. Here, we introduce KnottedGraph, a computational framework that converts such scientific representations to knotted graphs that retain graph connectivity and spatial embedding together. It constructs projected diagrams and PD codes, enabling various topological analyses, including Yamada-polynomial evaluation for topological classification. For scalable exact evaluation, it combines partial resolutions that leave the same unresolved connections and optimizes their processing order; the resulting algorithm is verified against published topological invariants of knotted graphs with up to 500 crossings. This scalability enables us to introduce an LLM-assisted mathematical-discovery methodology, in which computational topological data generated across knotted-graph families are used to identify candidate closed-form formulas. With this approach, we identify analytical Yamada-polynomials for generic graph motif families exhibiting Abelian and non-Abelian word sequences. Together, these scalable capabilities make knotted-graph topology computationally accessible across scientific domains, enabling large-scale classification and introducing a route from topological data to LLM-assisted AI4Math discovery.

• 24 wrz Kashti Satish Umare. A Concordance Invariant For Knots from $CFK^\infty$. arXiv PDF

  We construct a multi-filtered smooth knot concordance invariant, $β(t^1, a, t^2)$, derived from the knot Floer complex $CFK^\infty$. $β$ is a three-variable piecewise linear function defined on $(0,2) \times \mathbb{R} \times (0,2)$. We demonstrate that $β$ satisfies a subadditivity relation. We apply $β$ to prove concordance results for a certain family of L-space knots. We show that the positive cones spanned by Teragaito's knots $K_{1,n}$ and $K_{2,n}$ are disjoint in the smooth concordance group $\mathcal{C}$. For positive L-space knots, we show $β$ completely determines the Alexander polynomial and distinguishes knots with identical $Υ$ invariants.

• 24 wrz Joseph Breen; Alexander Zupan. Derivative links in contact topology. arXiv PDF

  We import the theory of $R$-links and derivative links into contact topology in both the Legendrian and transverse setting. This framework is used to characterize various forms of Lagrangian and symplectic sliceness, and more generally establishes one approach to what we call the Lagrangian slice-ribbon conjecture. Our main theorem asserts that a Legendrian knot with Thurston-Bennequin invariant $-1$ is Lagrangrian slice (resp. regularly Lagrangian slice) if and only if it supports a tight transverse (resp. Legendrian) $R$-link derivative.

• 24 wrz Butian Zhang. Pairings of combinatorial 1-cocycles with loops in knot spaces. arXiv PDF

  We show that the evaluations of combinatorial 1-cocycles defined by Gauss diagrams with a triangle on the canonical loops in the space of long knots are finite type invariants. Using Gauss diagrams, we construct two $\mathbb{Z}$-valued combinatorial 1-cocycles $β_1, β_2$ and a $\mathbb{Z}/2\mathbb{Z}$-valued 1-cocycle $β_3$ on the space of long knots and prove their cocyclicity by verifying their invariance under higher Reidemeister moves coming from the codimension-two singularities of plane curves. We show that they represent genuinely new 1-cohomology classes and compute their pairings with the rotation, rolling, half rolling, bracket and half bracket loops. A key new feature is that $β_1$ and $β_2$ can pair nontrivially with bracket and half-bracket loops. We conjecture that $(α_3^1,β_1,β_2)$ over $\mathbb{Q}$, and their mod 2 reductions together with $β_3$ over $\mathbb{Z}/2\mathbb{Z}$, form bases of degree-one cohomology up to order 4 in the sense of Vassiliev. In addition, we show that the reparametrization loop is homotopic to the rolling loop concatenated with the rotation loop in $\operatorname{Emb}(S^1, S^3)$. Finally, we give the criteria for a 1-cohomology class in the long knot space to descend to 1-cohomology classes in $\operatorname{Emb}(S^1, S^3)$ and $\operatorname{Emb}(S^1, S^3)/\operatorname{Diff}^{+}(S^1)$. Using these criteria, we show that $β_1$ descends to a nontrivial 1-cohomology class in $\operatorname{Emb}(S^1,S^3)$, while $β_1 \bmod 2$ and $β_3$ descend to linearly independent nontrivial 1-cohomology classes in $\operatorname{Emb}(S^1, S^3)/\operatorname{Diff}^{+}(S^1)$.

• 24 wrz Michal Jablonowski. Fibred realizations of the prism quandle $P_7$ in the standard four-sphere. arXiv PDF

  We study smooth fibred two-knots whose fundamental quandle is the prism quandle of order fifty-six. The candidates arise from framed circle surgery along a section of a mapping torus, using the two possible normal-framing classes. We prove that both ambient homotopy four-spheres are diffeomorphic to the standard four-sphere. The candidates have the same knot group and diffeomorphic exteriors, but they are inequivalent even as topological pairs. Every smooth fibred realization of this quandle in a homotopy four-sphere is equivalent, up to knot orientation, to exactly one of these two knots. Standardness follows from a Whitehead-link model for the orbit orbifold together with circle-action recognition. Inequivalence is detected by the third cyclic branched covers, on which a spin-bordism secondary invariant takes different values. Thus the parameter-seven prism quandle admits smooth fibred realizations in the standard four-sphere.

• 24 wrz Shreya Sinha; Allison Tsypin. Exotic knot traces and the epsilon-invariant. arXiv PDF

  Building on recent work of Hom and Wan, we construct additional examples of exotic knot traces, providing partial progress toward their conjecture. Our approach uses the immersed-curve techniques in Heegaard Floer homology developed by Chen and Hanselman.

• 24 wrz Qinghui Meng; Ximin Liu; Boxin Zhou. Quandle coloring quivers of pretzel links. arXiv PDF

  In this paper, we conduct a systematic study of quandle colorings and quandle coloring quivers for pretzel links using the dihedral quandle $\mathbb{Z}_{n}$. First, we systematically investigate all possible colorings of 3-pretzel links, determining the number of distinct colorings in each case as well as the structure of their quandle coloring quivers. In order to obtain more general conclusions, we impose restrictions on $n$ based on the properties of the coefficient matrix of the system of congruence equations. So we examine the number of quandle colorings and the quandle coloring quivers for 4-pretzel links in the case where $n$ is prime. Finally, Combining the results of 4-pretzel links we rigorously derive both the coloring numbers and quandle coloring quivers for general $m$-pretzel links in the case where $n$ is prime, with full proofs provided.

• 24 wrz Airi Aso. Asymptotic behavior of twisted Alexander invariants for hyperbolic knots with at most six crossings. arXiv PDF

  Let $K$ be a hyperbolic knot and let $ρ_n$ be the $n$-dimensional irreducible representation induced from a lift of its holonomy representation. Motivated by Goda's asymptotic volume formula and the complexified Volume Conjecture, we study whether the higher-dimensional twisted Alexander invariants associated with $ρ_n$ detect the complex volume of the knot complement. We compute $$ \fracπ2 \log \left( \fracA_K,n-2(1)A_K,n+2(1) A_K,n(1)^2 \right) $$ for all hyperbolic knots with at most six crossings. Our numerical experiments indicate that these values approach $$ \operatornameVol(S^3\setminus K) +i\,2π^2\operatornameCS(S^3\setminus K) $$ modulo $iπ^2\mathbb{Z}$. Based on these computations, we propose a complexified analogue of Goda's asymptotic volume formula.

• 24 wrz Aakash Gupta; Louis H. Kauffman; Rama Mishra. Projective Braids in $\mathbb{R}P^3$. arXiv PDF

  Links in projective space $\mathbb{R}P^3$ can be represented by diagrams in $\mathbb{R}P^2$ where the projective plane $\mathbb{R}P^2$ is represented by a disk with antipodal identifications on the boundary. In this context if we place an $n-$strand braid $β$ on this disk keeping its end points on the boundary then due to the identification of antipodal points it naturally represents a link diagram in $\mathbb{R}P^3$. We call this the projective closure of the braid $β$. In this paper we show that not all links in $\mathbb{R}P^3$ possess a diagram represented by projective closure of some braid.

• 24 wrz Keisuke Himeno; Masakazu Teragaito. Asymmetric L-space knots with arbitrary braid index. arXiv PDF

  The first examples of asymmetric L-space knots were found by Baker and Luecke. Among Baker-Luecke knots, the simplest one has braid index 12. Later, it turned out that there are just 9 asymmetric hyperbolic L-space knots in the SnapPy census, and their braid indices take the values 4, 5, 6 and 7. It is known that asymmetric L-space knots have braid index at least 4. Recently, Baker and the second author give infinitely many asymmetric hyperbolic L-space knots with braid index 4. In this paper, we construct an asymmetric hyperbolic L-space knot with arbitrary braid index bigger than 4. In fact, there exist infinitely many such knots for each braid index.

• 23 wrz Tatsuya Miura. Milnor's inequality and circular elastic knots. arXiv PDF

  We establish the extension of Milnor's inequality, relating total curvature with the bridge index, to the $C^1$-closure of a knot class, without any restriction on the self-intersections of the limit curve. Together with recent work of Reiter–von der Mosel, this resolves the circular elastic knot conjecture, first predicted by Gallotti–Pierre-Louis in 2007 and then formulated as a mathematical conjecture by Gerlach–Reiter–von der Mosel in 2017. More precisely, if the bridge and braid indices of a tame knot class coincide, then the multiply covered circle is the unique elastic knot.

• 23 wrz Toshiki Matsusaka. Modular integrals for individual modular knots. arXiv PDF

  To each modular knot, we attach a weight $2$ modular integral for $\mathrm{SL}_2(\mathbb{Z})$ whose homogenized cycle integrals recover its linking numbers with other modular knots in $S^3$. This construction answers Choie-Zagier's question of explicitly constructing modular integrals with prescribed rational period functions in weight $2$. Our formula applies to individual modular knots without symmetrization, thereby realizing an approach discussed but not pursued by Duke-Imamoglu-Tóth. This gives an answer to Ghys' question on pairwise linking numbers in terms of modular integrals, as sought by Simon.

• 22 wrz Thomas C. Martinez; Matthew J. Tyler. Toric Richardson Varieties and Slice Links. arXiv PDF

  For type $A$ braid varieties, including Richardson varieties and positroid varieties, we prove that the standard torus has a dense orbit exactly when the associated link is smoothly slice. More generally, we relate the smooth slice genus to the codimension of a generic standard-torus orbit. For knots, this codimension equals twice the slice genus. We also characterize algebraic tori among open affine Richardson varieties and positroid patches by 2-crown avoidance in their Bruhat intervals. Finally, positroid links have equal Seifert and slice genera, so a positroid link is smoothly slice exactly when it is an unlink.

• 22 wrz Thiago de Paiva; Yi Liu. Every Link Has Infinitely Many Explicit Generalised T-Link Presentations. arXiv PDF

  Generalised $T$-links provide a simple description of all links in $S^3$ as closures of products of standard twisting blocks, parametrised by finite sequences of integers. We prove that every link admits infinitely many pairwise distinct generalised $T$-link presentations. Starting from any such presentation, we give explicit parameter transformations that preserve the represented link and generate families of pairwise distinct presentations depending on arbitrarily many independent integer parameters.

• 21 wrz Joseph Geisz; Chris Peterson; Clayton Shonkwiler. Trefoil Probabilities and Polyhedra. arXiv PDF

  We determine the exact probability that the closed hexagonal polygon obtained by cyclically joining six independent points uniformly distributed on the unit sphere is knotted. The only possible nontrivial knot is a trefoil. Almost surely, the convex hull of the six points has one of two simplicial combinatorial types: a combinatorially regular octahedron and a combinatorially non-regular octahedron. We show that the straight-line complete graph on the six vertices of the hull of the combinatorially regular octahedral type admits exactly one unoriented trefoil Hamiltonian cycle. On the other hand, no cycle connecting vertices of the irregular octahedron can produce a trefoil. We then use stereographic projection to transform the probability of the regular hull type to a Sylvester-type problem for the planar beta-prime probability measure $dμ(x,y)=\frac{dx dy}{π (1+x^2+y^2)^2}$. Applying Stokes' theorem and the Blaschke-Petkantschin formula, we compute the expected squared $μ$-content of a random triangle. This yields a regular octahedral probability of $\frac{15}{4π^2}$ and a trefoil probability of $\frac{1}{16π^2}$.

• 21 wrz Malors Espinosa. Knots and the Sierpinski Tetrahedron. arXiv PDF

  In this paper we prove that there are infinitely many knots that cannot be embedded in the 1-skeletons of the finite iterations of the Sierpinski tetrahedron fractal. We do this by proving that such an embedding induces a sphere decomposition of weight at most 6. There are infinitely many knots with spherewidth greater than this.

• 21 wrz Yonghan Xiao. More Versions of Real Link Floer Homology. arXiv PDF

  In this paper, we further develop the real link Floer homology defined by the first author. We introduce a new base-pointing convention that leads to a different version of real link Floer homology and show that this new theory is related to the old one by an exact triangle. We also define a real link Floer theory for multi-based strongly invertible links, which is a strong real Heegaard invariant, and take a first step toward a real link Floer TQFT. A computer implementation for the new theory via grid diagrams was written by Zhenkun Li. We also include an appendix containing real grid homology of more than 50 small knots.

• 20 wrz Adam Klukowski. Counting tangles in coverings of closed 2-orbifolds via ribbon categories. arXiv PDF

  We describe what a typical covering of a closed 2-orbifold looks like on a small scale. Specifically, we prove that a uniformly random covering of degree n contains in expectation $Θ(n^0)$ short closed geodesics, potentially $Θ(n^0)$ pairs of nearby order-2 cone points, in general $Θ(n^{\frac{1}{m}})$ cone points of order m, and with high probability no small regions with more complicated topology. This generalises many results of Magee and Puder (2023) from orientable surfaces to possibly non-orientable orbifolds, and answers some of the questions raised by Puder and Zimhoni (2024).

• 20 wrz Nicolle González; John Lentfer; Hanna Mularczyk. Sign components of diagonal superspace coinvariants. arXiv PDF

  We prove the sign-isotypic components of the coinvariant rings $R_n^{(2,1)}$ and $R_n^{(2,0)} \otimes R_n^{(0,1)}$ are isomorphic and show that the triply-graded multiplicity of this sign character is the Schröder polynomial $S_n(q,t,a)$, divided by $1+a$. This settles the sign-character component of a conjecture of Zabrocki (2019) on a module for the Delta theorem and proves a conjecture of F. Bergeron (2020) on the multiplicity of the sign character of $R_n^{(2,1)}$. Finally, using a result of Hogancamp (2017), we enhance a recent result of Gorsky–Mellit (2026) which relates the Khovanov–Rozansky homology of the $(n,n+1)$-torus knot to $R_n^{(2,0)} \otimes R_n^{(0,1)}$, by showing that the associated Poincaré series for this knot can be computed from the sign component of $R_n^{(2,1)}$.

• 19 wrz Mark Pencovitch. Simple Slice Surfaces. arXiv PDF

  Given a simply-connected $4$-manifold with boundary the $3$-sphere, we give sufficient conditions for a knot in the boundary to bound a simply embedded locally flat surface in the $4$-manifold of fixed nonzero genus, representing a fixed nonzero homology class. In certain indefinite cases, we also give conditions for these slice surfaces to be unique up to equivalence.

• 17 wrz Marco Golla; Juanita Pinzón-Caicedo. The Conway knot has infinite order in smooth concordance. arXiv PDF

  We prove that the Conway knot has infinite order in the concordance group, and in particular that it is not slice. Ours is an alternative, independent proof with respect to those of Piccirillo and of Donatone, Kegel, Lewark, and Truöl.

• 17 wrz Filipp Buryak. Representation stability of string links and manifold links. arXiv PDF

  We prove representation stability for rational cohomology of spaces of string links and manifold links in Euclidean space as the number of links grows. For string links we consider the component of the standard embedding of disjoint copies of $\mathbb{R}^{m}$ into $\mathbb{R}^{d}$, where $m \geq 1$ and $d \geq m+3$. For manifold links we consider the component determined by disjoint copies of a fixed embedding of a closed smooth $m$-manifold into $\mathbb{R}^{d-2}$, where $d \geq 3$. In both cases, the rational cohomology groups form finitely generated $\mathrm{FI}^{\#}$-modules in each degree with generation arity at most $n\frac{d-2}{d-m-2}$ in degree $n$. We also obtain explicit generation bounds for the rational homotopy groups. The proofs combine geometric constructions of structure maps, hairy graph complex models for embeddings modulo immersions and a functorial framework that allows generation bounds to be established after forgetting the symmetric group actions.

• 17 wrz Nancy Scherich; Nathaniel Song. Combinatorial Goussarov-Polyak-Viro Formulas for the Linking Number and Low Degree Coefficients of the Conway Polynomial. arXiv PDF

  The linking number and coefficients of the Conway polynomial are two examples of finite type invariants. Goussarov-Polyak-Viro proved that any finite type knot invariant can be computed in a two step process, where the second step is referred to as the GPV map for the invariant. This paper computes the GPV maps for the linking number and low-degree coefficients of the Conway Polynomial. Chmutov-Khoury-Rossi gave one description of the GPV maps for coefficients of the Conway polynomial in terms of arrow diagrams and state-sum calculations. We take a much more grounded approach using fundamental linear algebra to describe the GPV maps in terms of a basis for the vector space of Gauss diagrams.

• 17 wrz Sam Ketchell. The Crossing Number and Arc Index of Satellite Knots. arXiv PDF

  By considering arc presentations and grid diagrams of satellite knots, we prove that if K is a satellite knot with companion C and wrapping number w, then its crossing number satisfies $c(K) \geq \frac{2}{5}(w^2c(C))^{1/11}$. We also prove other similar bounds regarding the crossing number and arc index of satellite knots and links. This is partial progress on Problem 1.2 of the K3 Problem List.

• 16 wrz Javier Martínez-Aguinaga. Families of knots that cannot be made Legendrian parametrically. arXiv PDF

  The fact that every smooth knot type admits a Legendrian representative is a classical result in contact topology. However, the analogous surjectivity question was open at the parametric level. In this work we address the $n>1$ case. We prove that for every $n\geq 3$, every knot type $\mathcal K$, every Legendrian representative $\mathcal L$ and every formal Legendrian representative $\mathcal{FL}$, the associated group homomorphisms $π_n(\mathcal{L})\toπ_n(\mathcal{K})$ and $π_n(\mathcal{FL})\toπ_n(\mathcal{K})$ are never surjective. We then show that surjectivity at the $π_2$-level depends on the knot type. This work thus proves the presence of rigidity for parametric families at every higher homotopy level beyond $π_1$.

• 15 wrz Anar Akhmedov; Sümeyra Sakallı. Gurtas Lefschetz Fibrations, Rational Blowdowns, and Exotic Symplectic Four-Manifolds. arXiv PDF

  We study negative spheres obtained from exceptional sections of Gurtas Lefschetz fibrations. Starting with the known system of $4n$ disjoint $(-1)$-sections, we show that a marked double sum contains $4n$ symplectic $(-2)$-spheres and $2n$ smooth $(-4)$-spheres obtained by pairwise tubing. By carrying the same sections through a fourfold sum, we instead obtain $4n$ disjoint symplectic $(-4)$-spheres and hence symplectic rational blowdowns. We then consider the Lefschetz fibrations on knot-surgered elliptic surfaces. Their branched-cover description gives $2n$ branch sections together with simultaneous geometric duals arising from node resolution. In the knot-surgery double these sections glue to symplectic $(-4)$-spheres, while the duals make the complement of every subcollection simply connected. The resulting rational blowdowns provide simply connected exotic symplectic four-manifolds under the parity condition stated in the main theorem. We also record the boundary-multitwist relation determined by the Gurtas sections.

• 15 wrz Jonathan Hanselman. A curve-based survey of knot Floer homology and concordance invariants. arXiv PDF

  Knot Floer homology associates to a knot in $S^3$ a bigraded chain complex over $\mathbb{F}[W,Z]$, from which many classical and concordance invariants can be extracted. Recent work shows that this algebraic object can equivalently be represented by a decorated immersed multicurve in a marked surface. This survey explains the immersed curve interpretation of knot Floer homology, aided by many examples, and shows how several invariants arising from the knot Floer complex can be extracted from the corresponding immersed curves. The decorated multicurve associated to a knot has a distinguished curve component $γ_0$ and a distinguished connected component $Γ_0$, both of which are concordance invariants of the knot. We pay particular attention to these components and various numerical concordance invariants that can be extracted from them. We introduce new generalizations of the $V_s$ invariants, and we give a new curve-based description of the Upsilon invariant by showing that it is determined by generalized $V_s$ invariants.

• 15 wrz Colin McCulloch. Computations of Floer Lasagna Modules for Traces of Negative L-Space Knots. arXiv PDF

  We show that for all negative $L$-space knots and framings $n\geq2g(K)$, if a certain cobordism map between cables of $K$ is non-vanishing, then the Floer lasagna module of the knot trace of $K$ with framing $-n$ is infinite dimensional. Further, for the maps on the hat flavour of link Floer homology induced by band maps, quasi-stabilisations and pair of pants cobordisms we give a description of the map on the $E^{\infty}$-page of Ozsvath and Szabo's spectral sequence for forgetting components.

• 15 wrz Anar Akhmedov. Xiao's Genus-Two Fibration: Branched Covers and Braid Monodromy. arXiv PDF

  We compute the geometric monodromy factorization of Xiao's genus-two Lefschetz fibration directly from its branched-cover construction, following Moishezon's braid-monodromy method. The complete quadrangle determines the motion of the six branch points and gives four nonseparating and three separating vanishing cycles. At each of the three $3+3$ degenerations, the branch motion determines the spherical mapping class, and Xiao's local holomorphic model determines its genus-two lift. After fixing a distinguished system of paths, we obtain an ordered positive factorization of type $(4,3)$ and give Artin coordinates for the seven factors.

• 14 wrz Thiago de Paiva; Connie On Yu Hui; José Andrés Rodríguez Migueles. The Lorenz braid index and hyperbolic volume. arXiv PDF

  A result of Futer, Kalfagianni, and Purcell implies that an upper volume bound for all link complements in the 3-sphere cannot depend solely on the braid index. In this paper, we introduce the Lorenz braid index and generalise the bunch algorithm to provide a general upper volume bound for all link complements in the 3-sphere. Such an upper bound is a quadratic polynomial in the Lorenz braid index. In addition, we construct an explicit family of hyperbolic Lorenz knots for which the classical braid index and the Seifert genus both tend to infinity, while the Lorenz braid index remains bounded.

• 13 wrz Leonardo Rodrigues de Medeiros; Arker Oke Soe. On Correspondences between the Alexander Polynomials of Special Alternating Links and MOY Graphs. arXiv PDF

  Fox's conjecture famously asserts that the absolute values of the coefficients of the Alexander polynomial of alternating links are trapezoidal. In the setting of MOY graphs, where a different notion of Alexander polynomial appears, the equivalent result to Fox's conjecture is known to hold. In this paper, we relate the two polynomials in the case of special alternating links. More precisely, we show that their degrees, first, and last coefficient agree, and that the MOY polynomial coefficients always dominate the classical Alexander polynomial.

• 13 wrz Anar Akhmedov. $\mathbb{Z}$-Torus Exteriors and Small Knot-Surgery Four-Manifolds. arXiv PDF

  We study a cut-and-paste operation in which torus neighborhoods in the author's symplectic building blocks $Y_K$ and $X_K$, associated to a genus-one fibered knot $K$, are replaced by marked exteriors of tori in $S^4$ with infinite cyclic complement group and two null peripheral slopes. We also recall the exact Luttinger-surgery realization of $M_K\times S^1$ from $Σ_g\times T^2$, keeping the knot-surgery/fiber-sum and Luttinger-surgery viewpoints in the same framework. For the trefoil block $Y_K$, two marked replacements give a simply connected manifold with intersection form $H$, hence a manifold homeomorphic to $S^2\times S^2$. A one-exterior gluing gives a smooth homotopy $4$-sphere. For the rank-six construction we use the identity double of two copies of $Y_K\setminusνΣ_2$, rather than the involutive gluing defining the original $X_K$. Two marked replacements along the surviving rim tori give a simply connected manifold with $e=8$ and $σ=0$. An explicit geometric basis has intersection form $3H$, so the resulting manifold is homeomorphic to $\#_3(S^2\times S^2)$. The Case II small-perturbation Seiberg–Witten invariant vanishes. The Seiberg–Witten invariant of the identity-glued rank-six family also vanishes; in particular, these manifolds are nonsymplectic.

• 13 wrz Kazumichi Nakamura. Delta-Unknotting Number for Montesinos Knots. arXiv PDF

  The $Δ$-unknotting number for a knot is defined as the minimum number of $Δ$-moves needed to deform the knot into the trivial knot. In this paper, we determine the $Δ$-unknotting numbers for certain families of Montesinos knots. Using results on pretzel knots and two-bridge knots, we prove that for these families the $Δ$-unknotting number equals the absolute value of the second coefficient of the Conway polynomial. In particular, every positive pretzel knot belongs to these families, and certain two-bridge knots also belong to them.

• 13 wrz Daren Chen; Ian Zemke; Hugo Zhou. Unknotting number and L-space satellite operations. arXiv PDF

  We give a lower bound for the unknotting number under the L-space satellite operations. In the case of braided L-space satellite operations, including cabling operations, we obtain a stronger bound. The proof uses a lower bound for the unknotting number in knot Floer homology due to Alishahi-Eftekhary. The computational tool is a formula for computing the knot Floer complex under the L-space satellite operations, which was defined in the authors' previous work.

• 13 wrz Lezhi Lin; Jonathan Spreer. Torus knots as loop-edges in three-sphere triangulations. arXiv PDF

  We present a procedure that, given a pair of coprime integers, produces a one-vertex triangulation of the three-sphere containing the corresponding torus knot as an edge. The construction is canonical, and produces a three-sphere triangulation from a pair of minimal layered triangulations of the solid torus. The size of the triangulation is, in the worst case, linear, but can be as small as logarithmic in the crossing number of the knot, motivating this triangulation size as an organising principle for knots, alternative to the crossing number.

• 12 wrz Chuwen Wang. Cyclotomic expansions of colored $SU(n)$ invariants of two-strand torus knots and Bailey transforms. arXiv PDF

  Habiro's cyclotomic expansion of the colored Jones polynomial has a higher rank analogue conjectured by Chen–Liu–Zhu for colored $SU(n)$ invariants. We prove this conjecture for torus knots $T(2,2p+1)$ and give a Bailey theoretic realization of the cyclotomic coefficients. For $n\geq2$ and $K_p=T(2,2p+1)$, the colored $SU(n)$ invariants admit an expansion $$ J_N^SU(n)(K_p;q) = \sum_m=0^N \left(\prod_j=0^m-1\N-j\\N+n+j\\right) H_m^(n,p)(q), $$ where $H_m^{(n,p)}(q)\in\mathbb Z[q^{\pm1}]$ is independent of the color $N$. The proof identifies the cyclotomic basis with a Newton basis and rewrites the resulting Newton coefficients as an ordinary Bailey transform. The Lin–Zheng formula for $T(2,2p+1)$ then gives a well-poised Bailey kernel. A terminating very-well-poised ${}_6φ_5$ summation diagonalizes the kernel and reduces the integrality problem to an ordinary Bailey transition. We prove a uniform integrality theorem for these transitions in a formal integral $q$-difference operator algebra. As a direct corollary, the expansion yields the corresponding congruence relations and proves part(i) of the Chen–Liu–Zhu $SU(n)$ volume conjecture for $T(2,2p+1)$.

• 11 wrz Hakan Akgün; Xianquan Yan; Ching Hua Lee. Topological classification through knotted graphs: Fermi surface dispersions and Lifshitz transitions. arXiv PDF

  Knot theory has provided a rich topological taxonomy for band structures, but its reach is fundamentally limited: knot invariants classify only 1D nodal lines at gap closure, and cannot encode the full dispersion or rich Fermi surface structure of realistic materials. Here we show that knotted graphs (knots that admit graph-like intersections in 3D space) - which have so far been elusive in condensed matter literature - provide a unified topological language for classifying the entire band dispersion, and even the eigenstate topology in some contexts. We propose a new framework beyond the existing Yamada polynomials that can topologically characterize the intricacies of realistic Fermi surfaces completely, crucially including how their multiple disconnected pieces are nested. This yields the Yamada set, a boundary-resolved extension which organizes the full topological evolution across energy into a Yamada sequence: a compact dispersion-level fingerprint directly tied to experimental signatures of Lifshitz transitions. Our framework is demonstrated with DFT-based band structures of real materials. Beyond dispersion-level classifications, this framework can be extended to non-Hermitian exceptional surfaces, where Berry-curvature flux further equips the knotted-graph skeleton with a directed Abelian edge flow that also captures the eigenstate topology.

• 11 wrz Rithwik Susheel Vidyarthi. Knot Floer homology of boundary Dehn twists. arXiv PDF

  We prove that the knot Floer complex, along with some restrictions on the flip map, determines the dual knot to $\pm 1$ surgery on the Borromean knot. In particular, this implies that Heegaard Floer homology detects boundary Dehn twists.

• 11 wrz Thomas Kindred. How natural of a geometric operation is Murasugi sum? arXiv PDF

  Gabai proved that any Murasugi sum of $π_1$-essential Seifert surfaces is also $π_1$-essential, and Ozawa extended this result to unoriented spanning surfaces. We show, however, that the analogous statement about geometrically essential surfaces is untrue. (A spanning surface is geometrically essential if it cannot be compressed or boundary compressed to another spanning surface.) We also ask when plumbing an unknotted annulus (with any number of twists) onto a compressible spanning surface yields a compressible surface. We provide positive and negative examples, and we establish a simple sufficient condition. As an application, we obtain a new constructive proof of a result of Hatcher and Thurston about essential spanning surfaces for 2-bridge knots and links.

• 11 wrz Davide Passaro; Lara San Martín Suárez. Quantum Invariants indexed by Fibered Faces of the Thurston Polytope. arXiv PDF

  We study the Gukov–Manolescu quantum invariant for oriented links. While this invariant is a single series in the knot case, we find that links admit multiple such series, each one consistent with the Melvin–Morton–Rozansky expansion of the colored Jones polynomials. We prove that convergent inverted state sums yield a family of multivariable Gukov–Manolescu series, indexed by monomials of the Alexander polynomial. We conjecture that these monomials correspond precisely to the fibered faces of the Thurston norm ball and provide extensive computational evidence for this correspondence. Further results concerning the leading term, the corresponding single-variable invariant, and the effect of partial Dehn surgery are also established.

• 10 wrz Hyungseok Jung. Amphichiral Knots: Odd Braid Index and Symmetry Classification in the Three-Braid Case. arXiv PDF

  We study how amphichirality of a knot constrains its braid index and, in the smallest nontrivial case, the combinatorics of its braid words. Using the Dynnikov–Prasolov resolution of the Jones conjecture, we show the braid index of an amphichiral knot is odd. This answers, in the negative, a question of Stoimenow on amphichiral knots of even braid index. We then classify prime amphichiral knots of braid index $3$. Every amphichiral knot of braid index $3$ is alternating and admits a minimal $3$-braid representative in a standard form encoded by a word $c$. Building on the Birman–Menasco classification of closed $3$-braids and the Murasugi normal form, we describe a dihedral action on $c$ under which the mirror and mirror-reverse operations are realized by a rotation and a reflection. For a standard form whose closure is a prime knot of braid index $3$, this yields a complete criterion: $\widehat{β_c}$ is amphichiral if and only if $c$ is a palindrome or has odd period, together with a determination of the precise symmetry type in terms of these properties and the presence of a non-degenerate flype.

• 10 wrz Chuanhuan Li; Ronggang Li. A Gap Theorem for the Möbius Cross Energy Near the Hopf Link. arXiv PDF

  We prove that the critical value $2π^2$ is isolated for the Möbius cross energy of two-component links in the round three dimensional sphere $\mathbb{S}^{3}$. Specifically, there exists $\varepsilon_0>0$ such that any non-split, regular $H^2$ pair of curves with disjoint images, having vanishing first variation and energy at most $2π^2+\varepsilon_0$, must lie in the Möbius-reparametrization orbit of the standard Hopf link.

• 10 wrz Ce Shen. The Chen-Yang volume conjecture for long integral fillings of fundamental shadow links. arXiv PDF

  We prove the Chen–Yang volume conjecture for all sufficiently long integral Dehn fillings of any fixed marked fundamental shadow-link exterior. For each fixed filling, the exponential growth of its $SO(3)$ Turaev–Viro invariants recovers its hyperbolic volume along the full sequence of odd levels. The filling coefficients may have mixed signs and unrelated magnitudes. We also establish a complete asymptotic expansion of the signed $SO(3)$ Witten–Reshetikhin–Turaev invariant and identify the absolute leading coefficient explicitly in terms of adjoint Reidemeister torsion. Fixed even colors on the filling cores recover characters of the geometric holonomy. The key difficulty is cancellation in the signed surgery sum. Our main analytic tool transfers an exact reflection symmetry from a continuous model to the finite quantum sums. We control the error below the exponential scale of the surviving contribution. We also apply the method to a one-edge state sum restricted to central colors. The dominant contributions cancel, and for each sufficiently large fixed number of blocks we determine the smaller surviving exponential rate and its nonzero leading coefficient.

• 10 wrz Anar Akhmedov. Braided Multisections and Symplectic Four-Manifolds with the Rational Cohomology of $S^2\times S^2$. arXiv PDF

  We construct symplectic four-manifolds by taking mixed fiber sums along explicit cyclic multisections in ruled surfaces. For a connected unbranched degree-$p$ multisection in $Σ_g\times S^2$, we determine the first homology and fundamental group of the complement and prove that its boundary is incompressible. It follows that no direct gluing of two such complements can be simply connected; moreover, the first homology of every direct sum retains finite quotients determined by the covering degrees. We classify the mixed sums having Euler characteristic $4$ and signature $0$. Up to interchanging the two summands, exactly three possibilities occur, corresponding to the degree pairs $(2,3)$, $(2,4)$, and $(3,3)$. For each of these cases, suitable adapted product-framed symplectic gluings have the rational cohomology ring of $S^2\times S^2$. Varying the gluing by symplectic transvections produces infinitely many pairwise nondiffeomorphic examples, distinguished by the unbounded orders of their finite first homology groups. We also construct the twisted ruled analogue of the $(2,4)$ case. Explicit finite-holonomy multisections give connected square-zero symplectic surfaces in the classes $2S_3-F_3$ and $4S_2-2F_2$ in the nontrivial $S^2$-bundles over $Σ_3$ and $Σ_2$, respectively. More generally, for a square-zero degree-$p$ multisection in the nontrivial bundle the complement has first homology $\mathbb Z^{2g}\oplus\mathbb Z/(p/2)$. Suitable gluings in the twisted $(2,4)$ case have $b_1=0$, $b_2=2$, and signature zero, and every such sum is non-spin. Hence they have the rational cohomology ring of $\mathbb CP^2\#\overline{\mathbb CP}^{\,2}$. We compare these constructions with the author's 2006 construction of minimal symplectic four-manifolds having the integral cohomology $S^2\times S^2$, obtained via knot surgery and twisted fiber sums.

• 10 wrz Noboru Ito. Homological lifts of Arnold invariants $J^-$ and $J^+$. arXiv PDF

  Viro's Euler-integral polynomial $P_C(q)$ and the Lanzat–Polyak quantized-curvature polynomial $I_q(C)$ refine Arnold's invariants $J^-$ and $J^+$ for generic immersed one-component plane curves. We construct homological lifts of both. The bigraded region homology retains the singular homology of every connected Alexander-index region; its graded Euler characteristic is $P_C(q)$. The triply graded smoothing-circle homology is generated by the oriented circles of the orientation-preserving smoothing and decategorifies to the smoothing term in $I_q(C)$. Keeping the actual region summands and the boundary regions of every smoothing circle gives a homological refinement of the oriented smoothing configuration, or Seifert state. An infinite family proves strictness: both polynomial data and the ordinary homological lifts agree, while the component-graded region homology and the branch-decomposed circle homology distinguish every pair. Further constructions recover the full $I_q(C)$ by a vertex complex, realize the local change of its curvature integral by edge homology, and give a canonical two-state homology for unoriented curves. Viro described his Euler-integral formula as an analogue of face state-sum formulas for quantum knot polynomials. Through the categorifications developed here, we obtain one concrete homological face-state-sum model realizing that analogy.

• 10 wrz Qiuyu Ren. A counterexample to the wrapping number conjecture. arXiv PDF

  We exhibit an annular knot with wrapping number four whose Kauffman bracket has annular degree at most two. This disproves the wrapping number conjecture.

• 10 wrz Greg Friedman; Nir Gadish; Robin Koytcheff; Dev Sinha; i in. Bar cohomology of links: beyond Milnor invariants. arXiv PDF

  We develop bar cohomology of link complements as an invariant of links in homology spheres. In this setting, bar cohomology is a Hopf algebra which is calculable using surfaces and their intersection curves in a link complement. In this first in a sequence of works, we introduce the invariant and show that it defines a canonical subspace of the tensor Hopf algebra, which already encodes information about Milnor's link invariants and provides geometrically significant information beyond them.

• 10 wrz Pedro Guicardi. On Diagrammatic Categorification of Verma Modules I: Braiding. arXiv PDF

  In this paper, we study the extensions of KLRW algebras to tensor products of Verma module representations of $\mathfrak{sl}_2$. Our motivation is to construct a theory of Khovanov homology for knot complements in $S^3$ (and which also categorifies the Gukov-Manolescu two-variable series for knot complements), which will be done in the second part of this work. We construct the categorification of R-matrices for Verma modules as functors given by derived tensor products with diagrammatic bimodules and explicitly compute their projective resolutions. We also prove these braiding functors induce an action of the braid group on the relevant categories. Then, we describe how to incorporate strands in finite-dimensional representations of $\mathfrak{sl}_2$, thereby establishing functors that serve as the Khovanov homology on a braid complement. In the case of the unknot, this gives knot homologies in $S^1\times D^2$, which we compare to Annular Khovanov Homology through several examples and show they are very closely related, conjecturing they are of the same dimension. We conclude with a proposal for the categorification of the cups and caps of Verma module colored strands, which we build upon in the next paper.

• 9 wrz Hongxu Chen. On Medial Quandle Coloring Link Invariant and Detecting Causality. arXiv PDF

  We investigate the capability of medial quandles to detect causality in (2+1) dimensional globally hyperbolic spacetime by determining if their coloring link invariants can distinguish between the connected sum of two Hopf links and an infinite series of relevant three-component links constructed by Allen and Swenberg in 2020, who suggested that any link invariant must be able to distinguish those links for them to detect causality in the given setting. We show that these quandles fail to do so as long as $a\sim b\Leftrightarrow a*b=a$ defines an equivalence relation. The Alexander quandles are an example to which this result applies. We also give a sufficient condition for a medial quandle to distinguish between the connected sum of two Hopf links and all links in Allen-Swenberg series, and give an example of such a quandle. Inspired by this result, we also derive a generalized theorem about the coloring of medial quandles on a specific type of tangle, which helps to determine whether these quandles can distinguish between a wider range of knots and links or not.