math.GT daily digest: 11 submissions for 1 October 2026

New submissions

The Thurston spine and equivariant cell decompositions in genus 2

• Authors: Ni An; Jiexing Chen; Ingrid Irmer; Paul Schmutz Schaller
• arXiv: 2609.38254
• Subjects: Geometric Topology (math.GT)

The Thurston spine of the genus 2 Teichmüller space of closed compact surfaces is obtained. In particular, we identify all mapping class group-orbits of filling sets of systoles in genus 2. Thereby, surprisingly rich cases already appear in this simplest case which is genus 2. This is contrasted with a cell decomposition of a bordification of Teichmüller space obtained computationally using Rivin's angle coordinates, on which the cells are labelled by Delaunay graphs closely related to our systolic graphs.

Khovanov Homology in Connected Sums, Properties and Applications

• Authors: Alan Du
• arXiv: 2609.38793
• Comments: 15 pages, 5 figures
• Subjects: Geometric Topology (math.GT)

We extend the definition of Khovanov-Lee homology to links in connected sums of interval bundles over surfaces and 's, and construct a Rasmussen-type invariant for links in these manifolds. As an application, we prove an inequality relating the Rasmussen-type invariant to the genus of surfaces with boundary in four-manifolds that are boundary connect sums of , , and .

Lens space surgeries on knots of small genus

• Authors: Misha Schmalian; Steven Sivek
• arXiv: 2609.39260
• Comments: 10 pages
• Subjects: Geometric Topology (math.GT)

We prove that the 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 , 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 $5$.

Montesinos Knots with Delta-Unknotting Number One

• Authors: Kazumichi Nakamura
• arXiv: 2609.39565
• Comments: 9 pages, 12 figures, 3 tables
• Subjects: Geometric Topology (math.GT)

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

The Euler classes of taut foliations on graph manifolds

• Authors: Yaoping Xie
• arXiv: 2609.39875
• Subjects: Geometric Topology (math.GT)

In this paper, we determine the set of Euler classes of all taut foliations for a family of closed graph manifolds. As a consequence, for any closed aspherical graph manifold $Y$, every integral class in the closed dual Thurston unit ball can be virtually realized as the Euler class of some taut foliation. In particular, $Y$ virtually admits a taut foliation with vanishing Euler class. We present examples of graph manifolds with arbitrarily large first Betti number, which admit no taut foliations with vanishing real Euler classes. This shows that the virtual requirement is necessary. We also provide examples of graph manifolds with arbitrarily large first Betti number, for which every taut foliation has vanishing real Euler class, but none has vanishing integral Euler class. This illustrates the subtle difference between real Euler class zero and integral Euler class zero.

Myopic Tutte polynomials and Khovanov homology in

• Authors: Keegan Boyle; Dean Spyropoulos
• arXiv: 2609.40122
• Comments: 43 pages, best viewed in color. Comments welcome!
• Subjects: Geometric Topology (math.GT); Combinatorics (math.CO)

We present a "myopic" Tutte polynomial for graphs on which takes only nullhomologous spanning subgraphs as input. It recovers the generalized Krushkal polynomial and Drobotukhina's analogue of the Jones polynomial for alternating, nullhomologous links in . We use this myopic Tutte polynomial to prove an analogue of the Kauffman-Murasugi-Thistlethwaite Theorem, relating the Jones polynomial of an alternating link to certain refinements of the crossing number. Finally, we construct a spanning tree model for the Khovanov homology of nullhomologous links, mirroring work by Champanerkar-Kofman and Wehrli for links in . For alternating links, we use our model to prove that the Khovanov homology in $\mathbb{Z}/2\mathbb{Z}$ coefficients is determined entirely by the Jones polynomial and signatures of the link.

Hierarchically Hyperbolic Surface-by-surface Groups

• Authors: Spencer Dowdall; Matthew Gentry Durham; Chris Leininger; Jacob Russell; Alessandro Sisto
• arXiv: 2609.40240
• Comments: 40 pages, 4 figures, no AI
• Subjects: Geometric Topology (math.GT)

We give many examples of surface bundles over surfaces whose fundamental groups are hierarchically hyperbolic by showing that extensions of surface groups constructed by the third author and Reid (LR surface groups) are hierarchically hyperbolic. This is facilitated by a new combination theorem for bundles of hyperbolic graphs over a hyperbolic base. This result differs from previous combination theorems for graph bundles by being applicable to bundles where the fibers are not properly embedded. We apply our combination theorem to establish the hyperbolicity of certain graph bundles that arise naturally from extensions of parabolically geometrically finite (PGF) subgroups of mapping class groups. These subgroups include LR surface groups, finitely generated Veech groups, free products of multi-twist groups, and many other examples created by a theorem of Udall. When the PGF groups have cyclic peripherals, hyperbolicity of our graph bundle is the key milestone towards showing that the extension groups are hierarchically hyperbolic.

Cross submissions

Complex structures on

• Authors: Philip Engel
• arXiv: 2609.38442
• Cross-list: math.AG
• Comments: 25 pages, 2 figures
• Subjects: Algebraic Geometry (math.AG); Complex Variables (math.CV); Differential Geometry (math.DG); Geometric Topology (math.GT)

A recent manuscript produced by Claude under the direction of Levent Alpöge constructs a 1-parameter family of compact complex threefolds diffeomorphic to . Our goal is to describe the geometric ideas behind the construction, give a self-contained proof, and remark on possible future research directions.

Intrinsic mirror symmetry for orbifold cubic surfaces and the spherical DAHA

• Authors: Pierrick Bousseau; Sayan Chattopadhyay
• arXiv: 2609.38556
• Cross-list: math.AG
• Comments: 28 pages, 8 figures. Comments welcome!
• Subjects: Algebraic Geometry (math.AG); High Energy Physics - Theory (hep-th); Geometric Topology (math.GT)

We determine the intrinsic mirror of an orbifold cubic surface with three nodes and interpret the resulting geometry through mirror symmetry between the and character varieties of the once-punctured torus. We further construct a quantization of the mirror using higher genus Gromov-Witten theory and show that it recovers the spherical double affine Hecke algebra.

Compact multi-monopole moduli spaces on Kähler surfaces

• Authors: Ollie Thakar
• arXiv: 2609.38595
• Cross-list: math.DG
• Comments: 29 pages
• Subjects: Differential Geometry (math.DG); Algebraic Geometry (math.AG); Complex Variables (math.CV); Geometric Topology (math.GT)

We consider the multi-spinor Seiberg-Witten equations on a 4-manifold in a general setting, where we allow the auxiliary bundle to be non-trivial. We show that when the metric is Kähler and the auxiliary connection is unobstructed anti-self-dual Yang-Mills, the moduli space of solutions is compact for all parameters in some open neighborhood of the given metric and auxiliary connection. We use this fact to compute a count of solutions on some examples of Kähler surfaces. These computations have a surprising purely algebro-geometric corollary which may be of independent interest.

Singularity of harmonic measures for hyperbolic lattices

• Authors: Nikolay Bogachev
• arXiv: 2609.40357
• Cross-list: math.DS
• Comments: 12 pages, comments are welcome
• Subjects: Dynamical Systems (math.DS); Group Theory (math.GR); Geometric Topology (math.GT); Probability (math.PR)

Let , , be a cocompact lattice containing a convex cocompact codimension-one subgroup in the sense of Sageev. We prove that every finitely supported admissible random walk on $\Gamma$ has hitting measure singular with respect to Lebesgue measure on . The proof extends the method of Kosenko--Tiozzo, replacing Fourier analysis for a cyclic subgroup by a von Neumann dimension argument that also applies to nonabelian subgroups. As corollaries, we prove the singularity conjecture of Kaimanovich and Le Prince for cocompact hyperbolic lattices admitting a proper cocompact cubulation, cocompact Kleinian groups, and cocompact hyperbolic lattices with totally geodesic sublattices of codimension one. In particular, our result covers all cocompact hyperbolic reflection groups, cocompact arithmetic lattices of simplest type and cocompact nonarithmetic lattices arising from the hybrid and inbreeding constructions.