math.GT daily digest: 7 new submissions for 13 August 2026

New submissions

Odd Vassiliev invariants vanish at four loops

• Authors: Greg Kuperberg
• arXiv: 2608.11334
• Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA)
• Comments: 10 pages

The odd Vassiliev conjecture asserts that all Jacobi diagrams with an odd number of legs vanish modulo the AS and IHX relations. Equivalently, the conjecture asserts that Vassiliev invariants cannot distinguish a knot from its inverse. The conjecture is elementary for one 1 or 2 loops, and was proven by Moskovich and Ohtsuki for 3 loops. We establish the conjecture for 4 loops. Our proof uses Jacobi diagrams and weight systems with polynomial coefficients, commuting colored legs, and several vanishing criteria.

Finiteness of singular orbits of pseudo-Anosov flows

• Authors: Lily Qiao Li
• arXiv: 2608.11377
• Subjects: Geometric Topology (math.GT); Dynamical Systems (math.DS)
• Comments: 10 pages, 6 figures, comments welcome!

We show that for any closed atoroidal 3-manifold $M$, there are only finitely many isotopy classes of links that can arise as singular orbits of pseudo-Anosov flows on $M$.

Spherical CR uniformizations of a sequence of hyperbolic 3-manifolds

• Authors: Jiming Ma; Baohua Xie
• arXiv: 2608.11571
• Subjects: Geometric Topology (math.GT)
• Comments: 41pages

Let $s782$ be the 2-cusped hyperbolic 3-manifold in the SnapPy census. Its spherical CR uniformization was established in \cite{JWX2023} using the Ford domain of the complex hyperbolic triangle group . By comparing the combinatorial structures of the Ford domain of and the Dirichlet domain of , we prove that for each , the Dehn filling of $s782$ along the slope on its second cusp admits a spherical CR uniformization, where denotes the meridian-longitude system of a cusp in SnapPy notation.

Construction and Periodicity of Minimal Resolution Graphs of Suspension Singularities

• Authors: Sümeyra Sakallı; Claudia Salles
• arXiv: 2608.11633
• Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG); Symplectic Geometry (math.SG)

We develop computational tools for constructing the minimal resolution graphs of suspension singularities. In particular, we provide a Magma program that implements Nemethi's algorithm that can be used to construct the minimal resolution graph of any suspension singularity. The program outputs the resulting resolution graph as structured graph data, encoding the vertices, edges, and decorations of the graph. We further develop a SageMath program that takes this data as input and produces graphical representations of the resolution graphs, facilitating their visualization and analysis. For three families of suspension singularities, we prove that the corresponding minimal resolution graphs are periodic, with periods equal to the orders of the corresponding monodromy factorizations. We also provide Magma programs for generating compactified resolution graphs of suspension singularities and checking graph isomorphisms over a specified range of parameters.

Quasisymmetric universality, quasi-isometric classification and topological rigidity of Kleinian groups

• Authors: Peter Haïssinsky; Yusheng Luo
• arXiv: 2608.12287
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR); Metric Geometry (math.MG)
• Comments: 87 pages, 24 figures

We study the quasisymmetric classification of limit sets of Kleinian groups and obtain a characterization of those geometrically finite limit sets that are quasisymmetrically universal. This allows us to obtain a quasi-isometric classification of finitely generated Kleinian groups. We also obtain a classification of virtually topologically rigid geometrically finite Kleinian groups.

Cross submissions

Decompositions of Tetrahedra into Height Functions

• Cross-list: Metric Geometry (math.MG)
• Authors: Connor Castillo; Jackson Smith
• arXiv: 2608.11217
• Subjects: Metric Geometry (math.MG); Combinatorics (math.CO); Geometric Topology (math.GT)

In short, we call a polyhedron P a height function if there exists some face F of P such that P lies entirely within F when orthogonally projected onto the affine plane containing F. Because tetrahedra come in various configurations, we propose a dihedral angle classification of tetrahedra, and prove that a tetrahedron T is either a height function already, or, at most one planar bisection of T is needed to produce two height function tetrahedra. Furthermore, we prove that such bisections belong to infinitely large families of bisections.

Nielsen classes in outer automorphism groups of free groups

• Cross-list: Group Theory (math.GR)
• Authors: Ilya Kapovich
• arXiv: 2608.12201
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT)

For every , we construct explicit families of generating pairs of and representing countably infinitely many Nielsen equivalence classes. These pairs arise as the homology images of explicit torsion generating pairs in and its index-two subgroup , yielding countably infinitely many Nielsen classes of generating pairs in both outer groups. Within each constructed outer family, all pairs become Nielsen equivalent after one stabilization, obtained by adjoining a trivial coordinate. We derive a relation-module obstruction to Nielsen equivalence of once-stabilized generating tuples. Evans's non-cancellation examples show that this obstruction is nontrivial and can distinguish Nielsen classes of once-stabilized generating tuples. We also record a quotient relation-module refinement for possible future applications.