math.GT daily digest: 8 new submissions for 31 July 2026

New submissions

A note on longitudes of virtual knots

• Authors: Lorenzo Traldi; Daniel S. Silver
• arXiv: 2607.27425
• Subjects: Geometric Topology (math.GT)
• Comments: 5 pages, 1 figure

It is a famous property that a longitude of a classical knot lies in the second commutator subgroup of the knot group. We observe that the same property holds for a longitude of a virtual knot.

An embedding

• Authors: Clayton McDonald
• arXiv: 2607.27619
• Subjects: Geometric Topology (math.GT)
• Comments: 1 page

In this note, we give a positive answer to K3 problem 4.28b, giving the first examples of homology 3-spheres that embed smoothly in but not .

Exotic knottings and symmetries of surfaces in 4-manifolds

• Authors: R. Inanc Baykur; Nathan Sunukjian
• arXiv: 2607.27751
• Subjects: Geometric Topology (math.GT); General Topology (math.GN)
• Comments: 23 pages

We study exotic knottings of surfaces in 4-manifolds through their ambient symmetries. We first give a general recipe for producing projectively rigid surfaces, for which every smoothly extendable self-diffeomorphism acts on first homology by plus or minus the identity. For every integer g >0, a refinement of this construction yields a finite sequence of genus-g surfaces F_0, ..., F_2g contained in a 4-manifold X_g. These surfaces are topologically isotopic and topologically flexible: every orientation-preserving self-diffeomorphism of F_i can be realized by a self-homeomorphism of X_g preserving F_i. Successive knotting, however, rules out increasingly many projective homological symmetries, revealing a finer knottedness phenomenon. The first two constructions combine iterated rim surgery with the convex geometry of Newton polytopes of relative Seiberg-Witten invariants. We also use hyperbolic geometry to construct a totally geodesic surface of positive genus whose smooth and topological extendable mapping class groups are both trivial.

Thompson's Group and Virtual Link Theory

• Authors: Micah Chrisman; Louisa Liles; Melody Molander
• arXiv: 2607.28406
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR); Quantum Algebra (math.QA)
• Comments: 36 pages, 41 figures

Thompson's groups were introduced in 1965 and have since found widespread application in fields as diverse as logic, group theory, homotopy theory, and lattice gauge theory. In 2014, V. F. R. Jones constructed unitary representations of , factoring through a surjection from to isotopy classes of links in . The second author extended Jones' surjection to , thereby constructing all isotopy classes of checkerboard colorable (CC) links in the thickened annulus. We complete this program for , defining a surjection from to virtual equivalence classes of CC links in thickened compact oriented surfaces. This yields a new oriented subgroup containing Jones' oriented subgroups and . We prove realizes all oriented almost classical virtual links. We then construct unitary representations of and from kei and operator quandle coloring invariants, respectively.

Path to homology of Yang-Baxter operators

• Authors: Jozef H. Przytycki
• arXiv: 2607.28626
• Subjects: Geometric Topology (math.GT); History and Overview (math.HO)
• Comments: 49 pages, 32 figures

This paper is an extended version of two talks I gave during workshop ``Loops'13" in Bedlewo in June this http URL the first talk I gave a historical introduction to Knot Theory. In the second, I traced my journey toward Yang-Baxter homology and this talk has a partially survey and a partially novel character.

Cross submissions

New relations for the vertex polynomial

• Cross-list: math.CO
• Authors: Scott Baldridge; Ben McCarty
• arXiv: 2607.27488
• Subjects: Combinatorics (math.CO); Geometric Topology (math.GT)
• Comments: 4 pages

We extend the vertex polynomial to graphs of arbitrary degree and prove local relations that hold when a graph contains a digon, triangle, quadrilateral or pentagon.

Realizing additive monoids as mapping degree sets

• Cross-list: math.AT
• Authors: Cristina Costoya; Vicente Muñoz; Bruno Valverde-Morales; Antonio Viruel
• arXiv: 2607.27993
• Subjects: Algebraic Topology (math.AT); Geometric Topology (math.GT)
• Comments: 15 pages, no figures

We prove that mapping degree sets are stable under multiplication by finite subsets of containing and by sets obtained from additive submonoids of through finitely many sums and products. In particular, every set of the latter type occurs as a mapping degree set. As a consequence, we obtain a broad family of infinite mapping degree sets, including finite unions of arithmetic progressions starting at . These results extend previous work on the realization problem and are related to a question posed by Neofytidis, Wang, and Wang.

Hyperbolicity of complements of orbits in Anosov flows

• Cross-list: math.DS
• Authors: Sergio Fenley; Tali Pinsky; Mario Shannon
• arXiv: 2607.28139
• Subjects: Dynamical Systems (math.DS); Geometric Topology (math.GT)
• Comments: 26 pages, 5 figures

We show that if an Anosov flow on a 3-dimensional manifold has orientable stable and unstable foliations, then the complement of any filling periodic orbit is a hyperbolic manifold. This generalizes the known case of the complement of a closed, filling geodesic orbit in the unit tangent bundle of a hyperbolic surface.

Furthermore, we show that the orientability condition on invariant foliations is necessary, by constructing a counterexample in the absence of this property.

In the case of the suspension flows we obtain that the complement of every collection of periodic orbits is hyperbolic, while for the geodesic flow (regardless of orientability of the invariant foliations) the complement of every filling and anannular collection of periodic orbits is hyperbolic.