math.GT daily digest: 6 submissions for 7 September 2026

New submissions

Minimal-Volume Equiangular Hyperbolic $4$-Polytopes

• Authors: Andrey Egorov
• arXiv: 2609.04258
• Subjects: Geometric Topology (math.GT)

We study finite-volume convex hyperbolic $4$-polytopes whose dihedral angles are all equal to a fixed strictly acute angle $\alpha$. Put and . We first show that the class is empty for . For , the unique polytope of minimum volume is the regular hyperbolic $4$-simplex with dihedral angle $\alpha$. As $\alpha$ increases to , this simplex degenerates to a Euclidean one and no hyperbolic simplex exists at . For , the unique minimum is the regular equiangular hyperbolic $4$-cube. The proof combines the hyperbolic Gram--Euler relation with the Davis--Okun theorem on the Charney--Davis inequality for flag triangulations of the $3$-sphere. The geometric step is a missing-face argument showing that the boundary of the dual of every nonsimplex in the class is flag.

A topological version of the Cartan-Hadamard Theorem and asphericity complexity

• Authors: Elias Gabriel Minian; Juan Martín Perez Garber
• arXiv: 2609.04440
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT)
• Comments: 15 pages

We investigate a topological version of the Cartan-Hadamard theorem that allows one to study asphericity of spaces via distinguished families of paths. A topological space has the distinguished path property (dpp) if there exists a continuous map from the space $\Pi (X)$ of homotopy classes of paths (relative endpoints) to the path space that is a right inverse to the canonical quotient map. For complete metric spaces endowed with a locally convex metric, the distinguished paths are precisely the local geodesics. We show that if $X$ has the dpp, then its universal cover is contractible and, in particular, $X$ is aspherical. The space $\Pi (X)$ is a fiber bundle over $X$ whose fiber is the universal cover of $X$, and in the case of Riemannian manifolds of non-positive curvature it is naturally isomorphic to the tangent bundle. We prove that every aspherical CW-complex that is locally finite or countable has the dpp, and this allows us to reinterpret asphericity of CW-complexes in terms of the existence of continuous sections. We also define and study the notion of asphericity complexity of spaces by means of local sections from $\Pi (X)$ to and relate it to the concept of equivariant topological complexity introduced by Colman and Grant.

A welded extension of 4-equivalence

• Authors: Graff Emmanuel
• arXiv: 2609.04624
• Subjects: Geometric Topology (math.GT)
• Comments: 12 pages, 18 Figures

We introduce a local move defining a welded extension of 4-equivalence. It induces the same quandle relations as 4-equivalence, giving a common algebraic framework for both the classical and welded settings. We also develop a version of arrow calculus adapted to this welded extension and apply it to the study of virtual tangles. This leads to a finite classification of algebraic virtual tangles without closed components under the resulting equivalence relation. As an application, we prove that the closure of any such tangle can be reduced to the unknot, giving a partial positive answer to a welded analogue of Nakanishi's conjecture.

Presentations of skein algebras of genus 2

• Authors: Haimiao Chen
• arXiv: 2609.05032
• Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA)
• Comments: 27 pages, 29 figures

We give a presentation for the Kauffman bracket skein algebra of a closed or one-holed oriented surface of genus $2$. This is the first such result for genus greater than $1$.

Two-numbers and Euler characteristics for quandles

• Authors: Ryoya Kai; Akira Kubo; Hiroshi Tamaru
• arXiv: 2609.05187
• Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG)
• Comments: 15 pages, 3 figures

Quandles can be regarded as generalizations of symmetric spaces. In the theory of symmetric spaces developed by Chen and Nagano, there is an interesting relationship between the two-number and the Euler characteristic. The two-number is a Riemannian geometric invariant that can also be characterized in terms of point symmetries, whereas the Euler characteristic is a topological invariant. The aim of this paper is to initiate the study of quandle analogues of Chen--Nagano theory. In particular, we investigate relationships between the two-numbers and the Euler characteristics of quandles, and provide examples of finite quandles that either satisfy or fail to satisfy properties analogous to those of symmetric spaces. These examples are constructed from directed simple graphs labeled by abelian groups.

On the Mahler measure and root distribution of the $Q$-polynomial of links

• Authors: Kotaro Shoji
• arXiv: 2609.05200
• Subjects: Geometric Topology (math.GT)
• Comments: 14 pages, comments are welcome!

We study the roots and the Mahler measure of the $Q$-polynomial of links. We first consider links obtained by adding twists to a pair of parallel strands. We prove that the Mahler measure of the transformed $Q$-polynomial converges as the number of twists increases. We also show that all but a uniformly bounded number of distinct roots of the $Q$-polynomial approach the real interval $-2,2$. This behavior is different from that of the roots of the Jones polynomial under twisting. Numerical experiments on prime knots lead us to a conjecture about the roots of the transformed $Q$-polynomial of alternating knots. Finally, we compare real and unit-circle roots of the Alexander, Jones, and $Q$-polynomials, and give an infinite family of $2$-bridge links whose $Q$-polynomials have only real nonzero roots.