math.GT daily digest: 8 submissions for 3 September 2026

New submissions

Minimum $b\_2$ of symplectic fillings of lens spaces

• Authors: Antony T.H. Fung
• arXiv: 2609.01791
• Subjects: Geometric Topology (math.GT); Symplectic Geometry (math.SG)
• Comments: 16 pages

We classify lens spaces for which the canonical negative-definite plumbing has minimal $b\_2$ among all symplectic fillings of the standard contact structure. The classification is given by 4 infinite families of "forbidden subgraphs" in a manner similar to Aceto-McCoy-Park's work on smooth negative-definite fillings of lens spaces. In particular, no classification of this form can be given using a finite list of forbidden configurations, highlighting a difference between symplectic fillings and smooth negative-definite fillings of lens spaces.

The bounded area class of negatively curved surfaces

• Authors: Roberto Frigerio; Ervin Hadziosmanovic
• arXiv: 2609.01830
• Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG); Differential Geometry (math.DG)
• Comments: 20 pages, comments are welcome!

Let $S$ be an oriented surface, possibly of infinite type, endowed with a complete Riemannian metric with pinched negative curvature. We prove that the area form defines a non-trivial class in the second bounded cohomology group of $S$, unless $S$ is diffeomorphic to the disc or the cylinder. This is in sharp contrast with the $n$-dimensional case, , where the non-triviality of the volume form in bounded cohomology is related to the Cheeger constant of the manifold. We also discuss how the bounded area class depends on the metric: we prove that, for compact surfaces, it recognizes constant curvature metrics among pinched negatively curved ones, while, even in the case of surfaces of infinite type, it does not distinguish non-isometric hyperbolic structures. More precisely, when $S$ is compact we show that, among the negatively curved structures of fixed area, the ones with constant curvature provide the unique minimizers for the norm of the bounded area class.

From Bowditch question to Goldman conjecture for type-preserving representations

• Authors: Viraj Joshi; Inyoung Ryu
• arXiv: 2609.02065
• Subjects: Geometric Topology (math.GT)
• Comments: 54 pages, 16 figures, Comments are welcome!

For punctured surfaces of genus , we explore the dynamics of the mapping class group action on the relative $\mathrm{PSL}(2,\mathbb{R})$-character varieties of type-preserving representations. For such relative character varieties, Goldman's conjecture predicts that the mapping class group acts ergodically on their non-Teichmüller components. A related question of Bowditch asks whether every non-elementary type-preserving representation that is non-Fuchsian sends some non-peripheral simple closed curve to a non-hyperbolic element. We show that, on the components of the relative character variety indexed by fixed signs of images of peripheral elements and relative Euler classes non-extremal in the generalized Milnor-Wood inequality, an affirmative answer to Bowditch's question implies Goldman's conjecture.

A topological version of Huber's theorem

• Authors: Ara Basmajian; Hugo Parlier
• arXiv: 2609.02274
• Subjects: Geometric Topology (math.GT)
• Comments: 9 pages

Let $X$ be a closed hyperbolic surface. We prove that the number of topological types of primitive closed geodesics of length at most $L$ is asymptotic to \[ \frac{1}{|\Isom(X)|}\frac{e^L}{2L}. \] as $L$ grows. Thus Huber's asymptotic remains unchanged after quotienting by topological type, up to the finite symmetry factor coming from the isometry group of $X$.

Frobenius Alexander quandles, knot colorings, and isogeny classes of abelian varieties over finite fields

• Authors: WonTae Hwang
• arXiv: 2609.02278
• Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG)

We introduce the notion of Frobenius Alexander quandles associated with abelian varieties over finite fields, and study them from both knot-theoretic and arithmetic viewpoints. We determine exactly for which rational primes a knot admits a nonconstant coloring by these quandles, in terms of the resultant of its Alexander polynomial and the Frobenius characteristic polynomial of the abelian variety. This leads to the notion of persistent colorability and its characterization, as well as applications to fibered knots and knots of small genus. We also prove a rigidity result showing that the isomorphism class of a Frobenius Alexander quandle determines the similarity class of the Frobenius endomorphism on the corresponding torsion subgroup. As a consequence, one sufficiently large Frobenius Alexander quandle determines the characteristic polynomial of the Frobenius endomorphism, and hence, determines the isogeny class of the abelian variety over the given finite base field.

Circuit Decomposition for Triangulations of Surfaces

• Authors: Jens Harlander; Maizie Quatrone
• arXiv: 2609.02701
• Subjects: Geometric Topology (math.GT)

An Euler circuit of a graph is a closed path that visits every edge of the graph exactly once. Euler circuit and circuit decomposition problems can also be formulated for higher dimensional simplicial complexes. An Euler k-circuit in K is a cyclic sequence of vertices v_1...v_n such that every k+1 adjacent terms { v_i,v_{i+1},...,v_{i+k} } (indexed modulo n) form a k-simplex, and every k-simplex of K appears exactly once in the sequence v_1v_2...v_n(v_1v_2...v_k). We investigate the 2-circuit decomposition problem for triangulated closed compact surfaces. For an orientable triangulated surface we use interior angles of paths to define an obstruction that lives in the first cohomology of the surface. It vanishes if and only if the surface has a 2-circuit decomposition. We also show that a non-orientable triangulated surface has a 2-circuit decomposition if and only if its orientable 2-fold cover does.

On the length conjecture for twist knots

• Authors: Samuel Panitch; Mauricio Romo
• arXiv: 2609.02706
• Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA)
• Comments: 34 pages

Using the newly developed $3$d quantum trace map, we compile more evidence for the length conjecture, a refinement of the all-order volume conjecture that incorporates insertions of additional links in the skein module of the knot complement. In particular, we prove that the length conjecture holds up to first order for an infinite family of twist knots. Along the way, we form a conjecture that the $3$d quantum trace map behaves naturally with respect to Dehn filling.

Cross submissions

The Bonnet-Myers theorem on Finsler manifolds with integral weighted Ricci curvature bounds

• Authors: Xinyue Cheng; Liulin Liu
• arXiv: 2609.02686
• Cross-list from: Differential Geometry (math.DG)
• Subjects: Differential Geometry (math.DG); Geometric Topology (math.GT)
• Comments: 32 pages. Any comments and suggestions are warmly welcome

In this paper, we derive some new relative volume comparison theorems and Bishop-Gromov volume comparisons on Finsler metric measure manifolds, all of which are controlled by the integral weighted Ricci curvature. In particular, we establish a Bishop-Gromov volume comparison theorem for nonconcentric balls. Based on these, we prove a theorem of Bonnet-Myers type on Finsler metric measure manifolds with integral weighted Ricci curvature bounds.