math.GT daily digest: 10 submissions for 28 October 2026

The Wednesday, 28 October 2026 arXiv math.GT new-submissions listing contains 10 eligible papers: 8 new submissions and 2 cross submissions. Replacement submissions are excluded.

New submissions

Recognizing convergence actions as proper isometric actions on hyperbolic spaces

Authors: Lihuang Ding, Abdul Zalloum
arXiv: 2610.36211
Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

Relying merely on B.I. (a.k.a. Bowditch's intelligence), we show that whenever a group $G$ admits a convergence action on a compact perfect metrizable space $M$ , there is a hyperbolic space $X$ along with a proper isometric action of $G$ on $X$ inducing a $G$ -equivariant homeomorphism This answers a question by Misha Kapovich.

Official arXiv abstract page

Filling knots and essential geodesic systole

Authors: Christopher J. Leininger, Yandi Wu
arXiv: 2610.36259
Comments: 28 pages, 8 figures, comments welcome. No AI use
Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

We prove that every closed, orientable $3$ --manifold contains hyperbolic knots with arbitrarily large essential geodesic systole. That is, for any $3$ --manifold $M$ , and there is a hyperbolic knot such that every closed geodesic in the knot complement which is non-null-homotopic in $M$ has length at least $R$ . The construction also produces filling knots, in the sense of Freedman-Krushkal [FreeKrus2023]. This answers two questions of Biringer.

Official arXiv abstract page

The unknotting problem for surfaces in the 4-sphere

Authors: Cameron Gordon
arXiv: 2610.36331
Comments: 17 pages
Subjects: Geometric Topology (math.GT)

We show that there is no algorithm to decide whether or not a closed, connected, orientable, locally flat PL surface in of genus at least 18 (resp. 19) is TOP (resp. PL) unknotted. We prove analogous results for non-orientable surfaces, and for surfaces in any compact PL 4-manifold. We also show that there is an orientable surface in of genus 18 whose group has unsolvable word problem, and an orientable surface in of genus 40 whose group contains an isomorphic copy of every recursively presented group. Similar results hold for non-orientable surfaces.

Official arXiv abstract page

Homological lifts of Arnold strangeness

Authors: Noboru Ito
arXiv: 2610.36387
Comments: 21 pages, 6 figures
Subjects: Geometric Topology (math.GT)

We construct homological lifts of Arnold's strangeness $St$ for long and closed plane curves. For long curves, we introduce a new $q$ -variable quantization polynomial , an edge state sum, designed for categorification and construct a triply graded homology that recovers . We prove strictness by exhibiting long curves with the same number of crossings and the same (in particular, the same rotation number and $St$ ) that are distinguished by the homology. For closed curves, we introduce a corresponding new polynomial and construct an edge homology. In this case is recovered from a signed $q$ -character defined on the canonical cycles of the homology rather than directly from a graded rank. Again, we exhibit closed curves with the same number of crossings and the same (in particular, the same rotation number and $St$ ) that are distinguished by the homology.

Official arXiv abstract page

On the $d$ -invariants of Brieskorn spheres and $GL (2, Z)$

Authors: Shimal Harichurn, Isabella Khan, András Némethi, Josef Svoboda
arXiv: 2610.36472
Comments: 26 pages, 8 figures, comments welcome
Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG); Number Theory (math.NT)

We study the $d$ -invariant from Heegaard Floer homology for Brieskorn spheres . Using surgeries on two singular fibers, we establish new relations of the $d$ -invariants among various Brieskorn spheres. For , we show that $d$ never vanishes if $q / r$ lies in certain intervals on the rational line. Altogether, we completely classify Brieskorn spheres with $d = 0$ . The zero set is expressed using submonoids of $GL (2, Z)$ and the Stern--Brocot tree. As an application, we prove that most Brieskorn spheres do not bound a rational homology $4$ -ball.

Official arXiv abstract page

Three-manifolds that do not admit veering triangulations

Authors: Layne Hall
arXiv: 2610.37057
Comments: 33 pages, 19 figures. Comments welcome
Subjects: Geometric Topology (math.GT)

We show that if a three-manifold $M$ admits a veering triangulation, then the dual spine of every essential triangulation of $M$ can be smoothed to a branched surface. We use this criterion to identify a large collection of cusped hyperbolic three-manifolds that do not admit veering triangulations. We then use the same criterion to construct the first infinite family of cusped hyperbolic three-manifolds without veering triangulations.

Official arXiv abstract page

Systolic pants decompositions with polynomial growth

Authors: Tarik Aougab, Jonah Gaster, Nicholas Miller
arXiv: 2610.37551
Comments: 13 pages, 3 figures
Subjects: Geometric Topology (math.GT)

We investigate improved bounds on the kissing number of a hyperbolic surface under the assumption that the systole set contains pants decompositions with volume growth constraints. Namely, when the dual graph of the pants decomposition has polynomial volume growth, we obtain a power-savings improvement for the upper bound on the kissing number in terms of systolic length. We also construct super-exponentially many (in $g$ ) pants decompositions with exponential volume growth for which our techniques still imply stronger upper bounds for kissing numbers. The proofs use the technology of ``stem systems'' and ``tulips'' developed by the first and second author to study $1$ -systems of curves.

Official arXiv abstract page

On Khovanov local equivalence groups

Authors: Robert Lipshitz
arXiv: 2610.37595
Comments: 40 pages. Research largely carried out by Anthropic's Claude models; paper written by the author. See methodology section and AI disclosure
Subjects: Geometric Topology (math.GT)

Inspired by work in Seiberg--Witten and Heegaard Floer homology, Dunfield, Schütz, and the author constructed an abelian "local equivalence group" built from pairs of chain complexes, and a homomorphism from the smooth concordance group to this group that uses the odd Khovanov complex and the Bar-Natan deformation of the even Khovanov complex. Lewark independently gave a related local equivalence group and concordance invariant, using only the Bar-Natan deformation. In this paper, we construct subgroups of Lewark's local equivalence group and the kernel of the map from our local equivalence group to Lewark's, as well as direct summands of Lewark's group and this kernel, invisible to the $s$ -invariants. The construction gives at least six $Z$ -valued concordance homomorphisms linearly independent from $s$ , $τ$ , and Dai--Hom--Stoffregen--Truong's invariants.

Official arXiv abstract page

Cross submissions

Counting Markov triples and simple geodesics

Authors: Morten S. Risager
arXiv: 2610.36069
Cross-list: math.NT
Comments: 21 pages, 5 figures. All comments welcome
Subjects: Number Theory (math.NT); Geometric Topology (math.GT)

We give an asymptotic formula for the number of Markov triples with largest entry below $x$ . We show that the error term is $o (lo g x)$ , which improves upon the previous best known results due to McShane and Rivin. We give an explicit error based on classical zero-free regions of the Riemann zeta function, and a sharper estimate under the quasi-Riemann hypothesis. The proof uses the correspondence between Markov triples and simple closed geodesics on the modular punctured torus and a careful analysis of the McShane-Rivin unit ball.

Official arXiv abstract page

Localization of weighted Goldman volume on $SL (3, R)$ -Hitchin Component

Authors: Shiyu Cao
arXiv: 2610.37317
Cross-list: math.DG
Comments: Preliminary version; all comments are welcome
Subjects: Differential Geometry (math.DG); Algebraic Geometry (math.AG); Geometric Topology (math.GT)

We compute Goldman volume on the moduli space of convex real projective structures on a closed oriented surface of genus at least two, with exponential weight defined by the shifted Blaschke area. For every positive parameter, the integral converges and is a finite sum of inverse powers of that parameter. Its coefficients are intersection numbers involving the bundle of regular cubic differentials on the moduli stack of stable curves. The proof establishes absolute convergence and vanishing signed boundary integrals of the transgression under simultaneous plumbing degeneration. The pushforward of Goldman measure by the shifted Blaschke area therefore has a polynomial density on $[0, \infty)$ , and its sublevel volumes are polynomial in the threshold.

Official arXiv abstract page

The Wednesday, 28 October 2026 arXiv math.GT new-submissions listing contains 10 eligible papers: 8 new submissions and 2 cross submissions. Replacement submissions are excluded.

New submissions

Recognizing convergence actions as proper isometric actions on hyperbolic spaces

Authors: Lihuang Ding, Abdul Zalloum
arXiv: 2610.36211
Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

Relying merely on B.I. (a.k.a. Bowditch's intelligence), we show that whenever a group $G$ admits a convergence action on a compact perfect metrizable space $M$ , there is a hyperbolic space $X$ along with a proper isometric action of $G$ on $X$ inducing a $G$ -equivariant homeomorphism $f : M \to \partial X .$ This answers a question by Misha Kapovich.

Official arXiv abstract page

Filling knots and essential geodesic systole

Authors: Christopher J. Leininger, Yandi Wu
arXiv: 2610.36259
Comments: 28 pages, 8 figures, comments welcome. No AI use
Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

We prove that every closed, orientable $3$ --manifold contains hyperbolic knots with arbitrarily large essential geodesic systole. That is, for any $3$ --manifold $M$ , and $R > 0$ there is a hyperbolic knot $K \subset M$ such that every closed geodesic in the knot complement which is non-null-homotopic in $M$ has length at least $R$ . The construction also produces filling knots, in the sense of Freedman-Krushkal [FreeKrus2023]. This answers two questions of Biringer.

Official arXiv abstract page

The unknotting problem for surfaces in the 4-sphere

Authors: Cameron Gordon
arXiv: 2610.36331
Comments: 17 pages
Subjects: Geometric Topology (math.GT)

We show that there is no algorithm to decide whether or not a closed, connected, orientable, locally flat PL surface in $S^{4}$ of genus at least 18 (resp. 19) is TOP (resp. PL) unknotted. We prove analogous results for non-orientable surfaces, and for surfaces in any compact PL 4-manifold. We also show that there is an orientable surface in $S^{4}$ of genus 18 whose group has unsolvable word problem, and an orientable surface in $S^{4}$ of genus 40 whose group contains an isomorphic copy of every recursively presented group. Similar results hold for non-orientable surfaces.

Official arXiv abstract page

Homological lifts of Arnold strangeness

Authors: Noboru Ito
arXiv: 2610.36387
Comments: 21 pages, 6 figures
Subjects: Geometric Topology (math.GT)

We construct homological lifts of Arnold's strangeness $St$ for long and closed plane curves. For long curves, we introduce a new $q$ -variable quantization polynomial $St_{q}$ , an edge state sum, designed for categorification and construct a triply graded homology that recovers $St_{q}$ . We prove strictness by exhibiting long curves with the same number of crossings and the same $St_{q}$ (in particular, the same rotation number and $St$ ) that are distinguished by the homology. For closed curves, we introduce a corresponding new polynomial $St_{q}$ and construct an edge homology. In this case $St_{q}$ is recovered from a signed $q$ -character $ch_{q}$ defined on the canonical cycles of the homology rather than directly from a graded rank. Again, we exhibit closed curves with the same number of crossings and the same $St_{q}$ (in particular, the same rotation number and $St$ ) that are distinguished by the homology.

Official arXiv abstract page

On the $d$ -invariants of Brieskorn spheres and $GL (2, Z)$

Authors: Shimal Harichurn, Isabella Khan, András Némethi, Josef Svoboda
arXiv: 2610.36472
Comments: 26 pages, 8 figures, comments welcome
Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG); Number Theory (math.NT)

We study the $d$ -invariant from Heegaard Floer homology for Brieskorn spheres $Σ (p, q, r)$ . Using surgeries on two singular fibers, we establish new relations of the $d$ -invariants among various Brieskorn spheres. For $p = 2$ , we show that $d$ never vanishes if $q / r$ lies in certain intervals on the rational line. Altogether, we completely classify Brieskorn spheres $Σ (2, q, r)$ with $d = 0$ . The zero set is expressed using submonoids of $GL (2, Z)$ and the Stern--Brocot tree. As an application, we prove that most Brieskorn spheres $Σ (2, q, r)$ do not bound a rational homology $4$ -ball.

Official arXiv abstract page

Three-manifolds that do not admit veering triangulations

Authors: Layne Hall
arXiv: 2610.37057
Comments: 33 pages, 19 figures. Comments welcome
Subjects: Geometric Topology (math.GT)

We show that if a three-manifold $M$ admits a veering triangulation, then the dual spine of every essential triangulation of $M$ can be smoothed to a branched surface. We use this criterion to identify a large collection of cusped hyperbolic three-manifolds that do not admit veering triangulations. We then use the same criterion to construct the first infinite family of cusped hyperbolic three-manifolds without veering triangulations.

Official arXiv abstract page

Systolic pants decompositions with polynomial growth

Authors: Tarik Aougab, Jonah Gaster, Nicholas Miller
arXiv: 2610.37551
Comments: 13 pages, 3 figures
Subjects: Geometric Topology (math.GT)

We investigate improved bounds on the kissing number of a hyperbolic surface under the assumption that the systole set contains pants decompositions with volume growth constraints. Namely, when the dual graph of the pants decomposition has polynomial volume growth, we obtain a power-savings improvement for the upper bound on the kissing number in terms of systolic length. We also construct super-exponentially many (in $g$ ) pants decompositions with exponential volume growth for which our techniques still imply stronger upper bounds for kissing numbers. The proofs use the technology of ``stem systems'' and ``tulips'' developed by the first and second author to study $1$ -systems of curves.

Official arXiv abstract page

On Khovanov local equivalence groups

Authors: Robert Lipshitz
arXiv: 2610.37595
Comments: 40 pages. Research largely carried out by Anthropic's Claude models; paper written by the author. See methodology section and AI disclosure
Subjects: Geometric Topology (math.GT)

Inspired by work in Seiberg--Witten and Heegaard Floer homology, Dunfield, Schütz, and the author constructed an abelian "local equivalence group" built from pairs of chain complexes, and a homomorphism from the smooth concordance group to this group that uses the odd Khovanov complex and the Bar-Natan deformation of the even Khovanov complex. Lewark independently gave a related local equivalence group and concordance invariant, using only the Bar-Natan deformation. In this paper, we construct $(Z /2 Z)^{\infty}$ subgroups of Lewark's local equivalence group and the kernel of the map from our local equivalence group to Lewark's, as well as $Z^{\infty}$ direct summands of Lewark's group and this kernel, invisible to the $s$ -invariants. The construction gives at least six $Z$ -valued concordance homomorphisms linearly independent from $s$ , $τ$ , and Dai--Hom--Stoffregen--Truong's invariants.

Official arXiv abstract page

Cross submissions

Counting Markov triples and simple geodesics

Authors: Morten S. Risager
arXiv: 2610.36069
Cross-list: math.NT
Comments: 21 pages, 5 figures. All comments welcome
Subjects: Number Theory (math.NT); Geometric Topology (math.GT)

We give an asymptotic formula for the number of Markov triples with largest entry below $x$ . We show that the error term is $o (lo g x)$ , which improves upon the previous best known results due to McShane and Rivin. We give an explicit error based on classical zero-free regions of the Riemann zeta function, and a sharper estimate under the quasi-Riemann hypothesis. The proof uses the correspondence between Markov triples and simple closed geodesics on the modular punctured torus and a careful analysis of the McShane-Rivin unit ball.

Official arXiv abstract page

Localization of weighted Goldman volume on $SL (3, R)$ -Hitchin Component

Authors: Shiyu Cao
arXiv: 2610.37317
Cross-list: math.DG
Comments: Preliminary version; all comments are welcome
Subjects: Differential Geometry (math.DG); Algebraic Geometry (math.AG); Geometric Topology (math.GT)

We compute Goldman volume on the moduli space of convex real projective structures on a closed oriented surface of genus at least two, with exponential weight defined by the shifted Blaschke area. For every positive parameter, the integral converges and is a finite sum of inverse powers of that parameter. Its coefficients are intersection numbers involving the bundle of regular cubic differentials on the moduli stack of stable curves. The proof establishes absolute convergence and vanishing signed boundary integrals of the transgression under simultaneous plumbing degeneration. The pushforward of Goldman measure by the shifted Blaschke area therefore has a polynomial density on $[0, \infty)$ , and its sublevel volumes are polynomial in the threshold.

Official arXiv abstract page

math.GT daily digest: 10 submissions for 28 October 2026

New submissions

Recognizing convergence actions as proper isometric actions on hyperbolic spaces

Filling knots and essential geodesic systole

The unknotting problem for surfaces in the 4-sphere

Homological lifts of Arnold strangeness

On the $d$ -invariants of Brieskorn spheres and $GL (2, Z)$

Three-manifolds that do not admit veering triangulations

Systolic pants decompositions with polynomial growth

On Khovanov local equivalence groups

Cross submissions

Counting Markov triples and simple geodesics

Localization of weighted Goldman volume on $SL (3, R)$ -Hitchin Component

Related content

math.GT daily digest: 10 submissions for 28 October 2026

New submissions

Recognizing convergence actions as proper isometric actions on hyperbolic spaces

Filling knots and essential geodesic systole

The unknotting problem for surfaces in the 4-sphere

Homological lifts of Arnold strangeness

On the $d$ -invariants of Brieskorn spheres and $GL (2, Z)$

Three-manifolds that do not admit veering triangulations

Systolic pants decompositions with polynomial growth

On Khovanov local equivalence groups

Cross submissions

Counting Markov triples and simple geodesics

Localization of weighted Goldman volume on $SL (3, R)$ -Hitchin Component

Related content

New submissions

Recognizing convergence actions as proper isometric actions on hyperbolic spaces

Filling knots and essential geodesic systole

The unknotting problem for surfaces in the 4-sphere

Homological lifts of Arnold strangeness

On the d-invariants of Brieskorn spheres and GL(2,Z)

Three-manifolds that do not admit veering triangulations

Systolic pants decompositions with polynomial growth

On Khovanov local equivalence groups

Cross submissions

Counting Markov triples and simple geodesics

Localization of weighted Goldman volume on SL(3,R)-Hitchin Component

Related content

New submissions

Recognizing convergence actions as proper isometric actions on hyperbolic spaces

Filling knots and essential geodesic systole

The unknotting problem for surfaces in the 4-sphere

Homological lifts of Arnold strangeness

On the d-invariants of Brieskorn spheres and GL(2,Z)

Three-manifolds that do not admit veering triangulations

Systolic pants decompositions with polynomial growth

On Khovanov local equivalence groups

Cross submissions

Counting Markov triples and simple geodesics

Localization of weighted Goldman volume on SL(3,R)-Hitchin Component

Related content

On the $d$ -invariants of Brieskorn spheres and $GL (2, Z)$

Localization of weighted Goldman volume on $SL (3, R)$ -Hitchin Component

On the $d$ -invariants of Brieskorn spheres and $GL (2, Z)$

Localization of weighted Goldman volume on $SL (3, R)$ -Hitchin Component