math.GT daily digest: 3 new submissions for 17 August 2026

New submissions

The crossing numbers of knots and links via 4D topology

• Authors: Akio Kawauchi
• arXiv: 2608.13789
• Subjects: Geometric Topology (math.GT)

The diagrams of links (including knots) are characterized in terms of circular rigid disk-chord diagrams of their spun ribbon torus-links in the 4-sphere. As a result, the crossing numbers of links are equal to the chord indexes of their spun ribbon torus-links. By using this result, additivity on the crossing numbers of links under connected sums can be shown.

Minimal Equiangular Hyperbolic Polyhedra in the Tetrahedral Range

• Authors: Andrey Egorov
• arXiv: 2608.14223
• Subjects: Geometric Topology (math.GT)

We consider finite-volume convex hyperbolic polyhedra whose dihedral angles are all equal to a fixed number $\alpha$. Such non-obtuse equiangular polyhedra may exist only for . The endpoint cases of the minimal-volume problem are known: for , the minimum is attained by the ideal regular tetrahedron, while for , among right-angled polyhedra, it is attained by the triangular bipyramid $P(3,2)$, whose volume is Catalan's constant. We prove the corresponding statement in the tetrahedral range , where the regular hyperbolic tetrahedron with dihedral angle $\alpha$ exists. Namely, for every such $\alpha$, among all equiangular hyperbolic polyhedra with dihedral angle $\alpha$, the minimum volume is attained only by this tetrahedron. The proof combines Andreev's theorem, the Schläfli formula, Atkinson's decomposition into atoroidal and prismatic parts, explicit volume estimates for ordinary prisms and complete orthoschemes, and a direct equiangular version of Inoue's edge surgery.

Cross submissions

Extending Goldberg's Exact Sequence to Braid Groups of Graphs and Simplicial Complexes

• Cross-list: Algebraic Topology (math.AT)
• Authors: Byung Hee An
• arXiv: 2608.14350
• Subjects: Algebraic Topology (math.AT); Geometric Topology (math.GT)

For a finite connected simplicial complex $X$, the strand map , from to , sends a pure braid to the homotopy classes of its strands. A theorem of Goldberg (1973) computes its kernel when $X$ is a closed surface other than and : the kernel is the normal closure of the pure braids supported in an embedded disc. We extend this picture to arbitrary finite connected simplicial complexes. Call $X$ $\text{weakly Goldberg}$ if some contractible subcomplex realises Goldberg's description, , and $\text{Goldberg}$ if can moreover be chosen so that is injective. We prove that the strand map is surjective if and only if ; that $X$ is weakly Goldberg if and only if its free part is a forest; and that $X$ is Goldberg if and only if it admits an $\text{admissible tree}$ -- a maximal tree of a scaffold, compatible with the boundary and interior types of the attachments of the free part to the thick components. We also classify the complexes for which the kernel is trivial, settle the exceptional surfaces and , and obtain complete answers for manifolds and for graphs. The main tools are a graph-of-spaces decomposition of the configuration space at a point of $X$ and a resolution procedure reducing an arbitrary complex to a simple model.