math.GT daily digest: 11 submissions for 7 October 2026

New submissions

Biquandle-Based Invariants of Virtual Knotoids under Connected Sum

• Authors: Hamdi Kayaslan, Selçuk İlbeyli
• arXiv: 2610.07187
• Comments: 24 pages, 11 figures
• Subjects: Geometric Topology (math.GT); Combinatorics (math.CO); General Topology (math.GN)

In this paper, we study the behavior of biquandle-based invariants of virtual knotoids under their connected sum. We first show that the fundamental biquandle of the connected sum of two virtual knotoids is the pushout of a span in the category of biquandles. By applying the Hom functor to this pushout description, we obtain the correspondence between biquandle colorings of and compatible pairs of colorings of summands. This provides a categorical explanation of a known matrix product formula for biquandle counting matrices under connected sum.

We then study the behavior of biquandle virtual bracket invariants under connected sum. We show that, for each coloring of the connected sum corresponding to a compatible pair of colorings of the summands and , the normalized biquandle virtual bracket value factors as the product of the normalized values of the summands. Building on this, we obtain connected-sum formulas for the normalized multiset invariants defined by utilizing biquandle virtual brackets.

When the coefficient ring is a number ring, the normalized bracket multisets can be encoded by polynomials and matrices with polynomial entries. We introduce a product $\star$ on monomials and an induced matrix product $\odot$. We then show that the normalized biquandle virtual bracket matrices satisfy

The colored Jones polynomial does not determine the Links--Gould polynomial

• Authors: Stavros Garoufalidis, Shana Yunsheng Li
• arXiv: 2610.07198
• Comments: 10 pages. No AI was used
• Subjects: Geometric Topology (math.GT)

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 -dimensional irreducible representation.

Reducible handle additions along separating slopes

• Authors: Qilong Guo
• arXiv: 2610.08145
• Comments: The manuscript contains 9 pages and 1 figure
• Subjects: Geometric Topology (math.GT)

Let M be a compact orientable simple 3-manifold, and let F be a component of the boundary of M with genus at least two. Suppose that alpha and beta are separating slopes on F and that both M(alpha) and M(beta) are reducible. We prove that the distance between alpha and beta is zero. Thus the reducible-reducible case of the conjecture of Qiu and Zhang on separating degenerating slopes holds in arbitrary boundary genus.

Reduced even and odd Khovanov homology of positive and $0$-adequate links

• Authors: Naageswaran Manikandan
• arXiv: 2610.08583
• Comments: 26 pages
• Subjects: Geometric Topology (math.GT)

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 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.

Top coefficients of ADO invariants are Murasugi-sum multiplicative

• Authors: Daniel López Neumann, Roland van der Veen
• arXiv: 2610.08638
• Comments: 30 pages
• Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA)

The Akutsu-Deguchi-Ohtsuki (ADO) invariants of links are a non-semisimple version of the colored Jones polynomials. We prove that the top coefficients of the ADO invariants of links in are multiplicative under Murasugi-sum along connected minimal genus Seifert surfaces. More generally, for any simple complex Lie algebra $\mathfrak{g}$ of type ADE and any , we prove that the top coefficient of the $r$-th $\mathfrak{g}$-ADO invariant is multiplicative under Murasugi-sum. As a corollary, we get that all the $\mathfrak{g}$-ADO invariants of fibred links are $q$-monic, generalizing a previous result of the authors for . Moreover, whenever is ribbon, we find an expression for the top coefficient of the $r$-th $\mathfrak{g}$-ADO invariant in terms of Hopf algebra integrals of universal -invariants of bottom tangles.

Cross submissions

Spectral gaps of double covers of hyperbolic surfaces

• Authors: Lawford Hatcher, Bram Petri
• arXiv: 2610.07193
• Cross-list: math.SP
• Comments: 15 pages, 2 figures
• Subjects: Spectral Theory (math.SP); Differential Geometry (math.DG); General Topology (math.GN); Geometric Topology (math.GT)

We prove that, in most moduli spaces of orientable hyperbolic surfaces of finite area, there exist surfaces all of whose covers of degree two have a new small eigenvalue. On the other hand, we show that four punctured spheres always admit a double cover without new small eigenvalues and that there exist closed surfaces of any genus with the same property. We prove the last two of these results using a version of Cheeger's inequality adapted to double covers.

On open covers of aspherical spaces satisfying -constraints and bounded Adamson cohomology

• Authors: Arturo Espinosa Baro, Stephan Mescher
• arXiv: 2610.07932
• Cross-list: math.AT
• Comments: 51 pages
• Subjects: Algebraic Topology (math.AT); Group Theory (math.GR); Geometric Topology (math.GT)

Given a topological space $X$ and a family of subgroups $\mathcal{F}$ of , the $\mathcal{F}$-category of $X$ is given as one less than the cardinality of the smallest cover of $X$ by open subsets whose fundamental groups lie in $\mathcal{F}$. In this article we study $\mathcal{F}$-categories of aspherical spaces and obtain cohomological lower bounds, a maximality result and bounded cohomology classes whose vanishing properties are crucial for determining $\mathcal{F}$-categories. For this purpose, we study the bounded Adamson cohomology of a family of subgroups and discuss universality properties of bounded cohomology classes. We apply our techniques to derive some applications to the study of the monotonicity of $\mathcal{F}$-categories under degree-one maps between manifolds and to provide a counterexample to a question of Capovilla, Löh and Moraschini.

What a knot sees at a root of unity

• Authors: Ya.Kononov, A.Morozov
• arXiv: 2610.08242
• Cross-list: hep-th
• Comments: 24 pages
• Subjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Geometric Topology (math.GT)

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 is a primitive $m$-th root of unity, e.g. , these polynomials exhibit a universal (knot-independent) factorization: becomes the product of the polynomials for the $m$-core of the Young diagram $R$ and of one and the same factor for every $m$-ribbon of $R$. In all the examples which we computed this factor is just the special polynomial at . At factorization follows from the singularity of the cabled HOMFLY, which is the same for all representations. At the relevant singularity is in the projector, and what survives of it is a finite sum of braids -- 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.

An AI-Assisted Formalization of the Poincaré Conjecture

• Authors: Zhiyuan Zhang, Axel Delaval, Leheng Chen, Jinxuan Chen, Jie Xu, Yuxuan Liao, Jiedong Jiang, Chunlei Liu, Bin Dong
• arXiv: 2610.08329
• Cross-list: cs.AI
• Comments: 15 pages, 2 figures. Code: this https URL
• Subjects: Artificial Intelligence (cs.AI); Geometric Topology (math.GT)

We present an AI-assisted Lean 4 formalization of the Poincaré conjecture. The project began with limited reusable formal infrastructure for the geometric analysis behind the proof. To organize this work, we combined a proof blueprint prepared by mathematicians with explicit milestone statements. These milestones enabled parallel agent work and gave mathematicians clear points to locate blockers and provide effective mathematical guidance. Our analysis identifies the human interventions and organizational choices behind this workflow. The project provides a starting point toward reusable infrastructure for future formalization projects; such infrastructure, once developed, could eventually reduce the cost of verifying mathematical results in geometric analysis.

Nearly Lefschetz fibrations, Stein structures, and regular Lagrangians

• Authors: Joseph Breen, Agniva Roy, Luya Wang
• arXiv: 2610.08440
• Cross-list: math.SG
• Comments: 53 pages, many figures, comments welcome!
• Subjects: Symplectic Geometry (math.SG); Geometric Topology (math.GT)

We study Stein structures on nearly Lefschetz fibrations, with applications to symplectic and Lagrangian submanifolds in Weinstein domains. We prove that, up to Weinstein deformation equivalence, all Lagrangian disks with Legendrian boundary in $4$-dimensional Weinstein domains are regular in the sense of Eliashberg-Ganatra-Lazarev. This settles a Weinstein analogue of the nearby Lagrangian conjecture for two-dimensional Lagrangian disks, and resolves part of a Lagrangian analogue of the Slice-Ribbon conjecture. Along the way, we show that positive allowable nearly Lefschetz fibrations are supported by canonical Stein structures and prove a corresponding quasiflexibility result in the planar case. This yields a generalization of work of Boileau-Orevkov which may be of independent interest. Finally, we give an explicit algorithm producing a Weinstein Kirby diagram from the nearly Lefschetz fibration structure of a multisection complement.

Xiao's Degree-Four Fibration and the Polizzi Model

• Authors: Anar Akhmedov, Sümeyra Sakallı
• arXiv: 2610.08655
• Cross-list: math.AG
• Comments: 28 pages
• Subjects: Algebraic Geometry (math.AG); Geometric Topology (math.GT)

Let be the surface in Xiao's degree-four family identified by Polizzi as a smooth divisor in $E(3)$. Starting from Polizzi's branch model and Xiao's classification of the thirteen singular fibers, we derive the local spherical braids and their Picard--Lefschetz lifts, including the colored $3+3$ partitions at the seven reducible fibers. We identify the normalization of each bisection with $E/\{\pm 1\}$ and its degree-two ruling map with the quotient by the involution induced by translation by a nonzero two-torsion point. We prove , compute the elementary-divisor-$4$ Albanese kernel of a regular genus-two fiber, and show that the six nonseparating Picard--Lefschetz transformations represent the six cusps of $\Gamma (4)$. We also prove that the natural product tori near an elliptic section are nullhomologous and cannot lower below $2$ by torus surgery. As a separate application of the degree-three monodromy, we give a twisted-double construction of an exotic .