math.GT daily digest: 8 new submissions for 31 August 2026

New submissions

A normal form for the Thin Flat Surfaces category

• Authors: Jaime Herrera, Carmen Rovi
• arXiv: 2608.27771
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT); Category Theory (math.CT)

The category TFS of thin flat surfaces, introduced by Khovanov, Qi, and Rozansky, is a strict symmetric monoidal category whose morphisms are compact oriented surfaces with corners arising as neighbourhoods of immersed graphs in a strip. We prove two main results. First, a normal form theorem: every connected viewable tf-cobordism is equal, in TFS, to a canonical composite determined by its topological type. Second, a sufficiency theorem: we establish a list of relations in the TFS category which are sufficient, meaning that any two composites of generators representing the same tf-cobordism are related by a finite sequence of the listed relations. Together, these results give a complete generators-and-relations presentation of TFS.

Trisection invariants of 4-manifolds are uncomputable

• Authors: Nathan M. Dunfield, Marc Kegel, Shana Yunsheng Li, Qiuyu Ren
• arXiv: 2608.27811
• Subjects: Geometric Topology (math.GT)

We prove that two invariants of smooth 4-manifolds defined in terms of trisections are uncomputable: the trisection genus and the Kirby-Thompson L-invariant. That is, there does not exist an algorithm that takes as input a triangulated closed orientable 4-manifold and outputs one of these quantities. The result on the L-invariant resolves the second part of Problem 4.116 in the K3 problem list. In the same spirit, we show that the PL multisection genus of PL manifolds is uncomputable in dimensions at least four.

Instantons, indefinite 4-manifolds, and Dehn surgery

• Authors: Aliakbar Daemi, Xingpei Liu, Mike Miller Eismeier
• arXiv: 2608.27904
• Subjects: Geometric Topology (math.GT)

We prove that there exist hyperbolic integer homology spheres with arbitrarily large Dehn surgery number. Previously, no integer homology sphere was known to have a surgery number larger than $2$. Our approach uses Froyshov's invariant $q\_3$ of integer homology spheres, which is defined in terms of mod 2 instanton homology. We show that if is a cobordism between integer homology spheres with no $2$-torsion in its first homology, then . We also extend both $q\_3$ and the inequality to rational homology spheres.

An example of biquandle and an invariant of long virtual knots defined through it

• Authors: Nozomu Sekino
• arXiv: 2608.27934
• Subjects: Geometric Topology (math.GT)

We introduce an example of biquandle and define an invariant of long virtual knots through it. As an application, we give a necessary condition for two long virtual knots commuting.

Entropy and domination for quasi-Hitchin representations

• Authors: Pabitra Barman, Subhojoy Gupta
• arXiv: 2608.27939
• Subjects: Geometric Topology (math.GT)

Let $S$ be a closed oriented surface of genus . We consider an $n$-pleated representation obtained by bending a Hitchin representation along a maximal geodesic lamination. The space of such $n$-pleated representations was recently introduced by Maloni-Martone-Mazzoli-Zhang who provided a parametrization via shear-bend cocycles. Our first result is that $\rho \_0$ dominates $\rho$ in the Hilbert length spectrum and the translation-length spectrum; this generalizes our earlier result for finite laminations on punctured surfaces. Using this, we prove entropy rigidity results: namely, the Hilbert entropy of a quasi-Hitchin representation in the bending fiber is strictly greater than that of $\rho \_0$, and the same for the translation-length entropy in the case that $\rho \_0$ is $n$-Fuchsian. The proof involves analyzing the weighted planar networks for finite approximants of the monodromy matrix, and establishing a strict domination for \emph{most} curves using the equidistribution of closed geodesics in the unit tangent bundle of $S$.

An Alexander Polynomial Refinement for Alternating Links, with Trapezoidal Properties

• Authors: Joe Boninger
• arXiv: 2608.28484
• Subjects: Geometric Topology (math.GT); Combinatorics (math.CO)

We define an invariant of alternating links---a homogeneous, four-variable Laurent polynomial---that encodes the symmetrized Alexander polynomial, the signature, and other topological data. Along the way, we extend a spanning tree formulation of the Alexander polynomial due to Murasugi and Stoimenow from special alternating links to all alternating links. This project is motivated by Fox's trapezoidal conjecture; accordingly, we prove certain sequences associated to our invariant are trapezoidal for all alternating links. We also conjecture our polynomial has $M$-convex support, and that it satisfies symmetry and log-concavity properties. We prove a partial symmetry result.

Cross submissions

On 2-Sphere Bowditch Boundaries Attaining Conformal Dimension

• Authors: Abhijit Pal, Rana Sardar
• arXiv: 2608.28346
• Cross-list: math.GR
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT)

Bonk and Kleiner proved that if $G$ is a Gromov hyperbolic group whose boundary $\partial \_\infty G$ is homeomorphic to an Ahlfors $Q$-regular metric $2$-sphere $Z$, and the Ahlfors regular conformal dimension of $Z$ is attained and equal to $Q$, then $G$ acts discretely, cocompactly, and isometrically on . In this article, we extend the Bonk-Kleiner theorem to the setting of relatively hyperbolic groups. More precisely, we prove that if $(G,\mathcal{H})$ is a relatively hyperbolic group whose Bowditch boundary is homeomorphic to an Ahlfors $Q$-regular metric $2$-sphere $Z$, with the Ahlfors regular conformal dimension of $Z$ attained and equal to $Q$, then $G$ acts discretely and isometrically on , and every subgroup in $\mathcal{H}$ is virtually .

The Twist Conjecture and the Isomorphism Problem for Coxeter groups

• Authors: Elia Fioravanti
• arXiv: 2608.28348
• Cross-list: math.GR
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT); Representation Theory (math.RT)

We prove Mühlherr's Twist Conjecture: any two angle-compatible Coxeter generating sets of a Coxeter group differ by a finite sequence of elementary twists and a conjugation. Combined with earlier work of Howlett-Mühlherr and Marquis-Mühlherr, this completes the resolution of the Isomorphism Problem for Coxeter groups. A further consequence is that $\mathrm{Aut}(W)$ is finitely generated for every Coxeter group $W$, and there is an algorithm producing a finite set of generators for $\mathrm{Aut}(W)$ starting from any Coxeter matrix. Of the vast literature on the Twist Conjecture, we utilise only two results in an essential way: strong rigidity of $2$--spherical Coxeter systems, due to Caprace and Mühlherr, and the framework of markings and hierarchies developed by Caprace and Przytycki for the twist-rigid case. We also exploit in a fundamental way some soft ideas from JSJ theory and an observation of Mihalik-Tschantz on splittings of Coxeter groups. No form of AI was used in the writing of this manuscript, nor in the research that it presents.