math.GT daily digest: 9 submissions for 19 October 2026

New submissions

The Involution Rank of \MCG(Σ_3)! Is Four

• Authors: Qilong Guo, Dongqi Sun
• arXiv: 2610.22774
• Subjects: Geometric Topology (math.GT)

For the closed orientable surface , we prove [ \irank\MCG(\Sigma_3)=4.] Thus \MCG(\Sigma_3)! can be generated by four involutions, but not by three. This settles the genus-three case of Farb's problem on generation by involutions.

Regular Additive Scalar Invariants of Hyperbolic Spin Supertriangles

• Authors: Ilia Gaiur, Robert Penner
• arXiv: 2610.23209
• Subjects: Geometric Topology (math.GT)

We classify regular cyclic scalar invariants of non-degenerate hyperbolic spin supertriangles which are additive under geodesic subdivision. After fixing the spin lift, the common-first gauge (introduced in earlier work by the second-named author and further developed here) gives a complete $3|4$ modulus of such supertriangles.. The space of additive scalar invariants is proved to have superdimension $3|1$: ordinary hyperbolic area in degree zero, two even generators in degree two, and one odd generator in degree three; the degree-one and degree-four sectors vanish. One degree-two generator is the fermionic first variation of body area, equivalently the nilpotent correction from ordinary body angle defect to super angle defect. The degree-three generator changes sign under reversal of the spin lift, while the other generators are fixed, so ordinary supertriangles carry only three even invariants.

Phases in the cyclic quantum dilogarithm: correction to "Cyclic quantum Teichmüller theory"

• Authors: Tsukasa Ishibashi
• arXiv: 2610.23572
• Comments: 20 pages, 1 figure
• Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA); Representation Theory (math.RT)

We first correct -phase issues in the basic definition around the cyclic quantum dilogarithm operators in [Ish26]. In particular, the notion of global coefficients must be extended by incorporating additional lifting data forming a -cover, and the pentagon relation generally acquires an extra -factor. Nevertheless, we prove that every point in a regular locus of the partially compactified cluster Poisson variety lifts to a \emph{strict} extended global coefficient, for which this extra -factor is trivial. Consequently, the original statements of [Ish26] hold for such lifts and are therefore restored.

Arithmeticity, superrigidity, and totally geodesic submanifolds of infinite-volume hyperbolic manifolds

• Authors: Nicholas Miller
• arXiv: 2610.23594
• Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG); Dynamical Systems (math.DS); Group Theory (math.GR)

Given an infinite-volume convex cocompact hyperbolic manifold with Fuchsian ends and infinitely many maximal properly immersed totally geodesic submanifolds of dimension at least two intersecting its convex core, we show that it (infinitely) covers an arithmetic hyperbolic manifold. This answers the general version of a question of Oh. As a byproduct of our techniques, we also give a Ratner-type classification of -invariant Radon measures whose support intersects the frame bundle of the interior of the convex core.

A Topological Quantum Field Theory for Representation Varieties of Torus Links

• Authors: Enrique Aycart-Maldonado, Alejandro Calleja
• arXiv: 2610.23644
• Comments: 32 pages, 5 figures. Comments are welcome
• Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG)

We construct a Topological Quantum Field Theory (TQFT) that computes the virtual classes of $G$-representation varieties associated with the fundamental group of a torus link complement. Following the framework established by González-Prieto, Logares and Muñoz for $G$-representation varieties of surface groups, we introduce a suitable bordism whose iteration enables a recursive topological evaluation of these virtual classes. Via this TQFT, we explicitly compute the virtual class of these varieties for the groups and .

Finite Cyclic Fundamental Groups and Exotic Universal Covers from the Cartwright--Steger Surface

• Authors: Anar Akhmedov, B. Doug Park, Sümeyra Sakallı, Sai-Kee Yeung
• arXiv: 2610.23711
• Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG); Symplectic Geometry (math.SG)

We use a particular nodal curve on the Cartwright-Steger surface to construct minimal symplectic four-manifolds with arbitrary nontrivial finite cyclic fundamental group and fixed topological invariants . Using these symplectic four-manifolds as building blocks, we construct a symplectic 4-manifold that is homeomorphic to for any integer .

Two-bridge knot epimorphisms and intersections in the character variety

• Authors: Isidora Bailly-Hall, Michelle Chu, Karina Dovgodko, Akash Ganguly, Jiachen Kang, Jishi Sun
• arXiv: 2610.23731
• Comments: 21 pages
• Subjects: Geometric Topology (math.GT)

We study the SL(2,C) character varieties of hyperbolic two-bridge knot groups admitting epimorphisms onto the trefoil knot group. Ohtsuki, Riley, and Sakuma give a systematic construction of an infinite family of such knots. The character varieties of these knot groups contain a component of irreducible representations arising from the epimorphism, which is distinct from the canonical component. We prove that every affine point of intersection between this component and any other component is algebraic non-integral, and therefore detects an essential surface in the knot complement. We also determine the boundary slope of each detected surface explicitly in terms of the continued fraction expansion of the two-bridge knot.

Cross submissions

Gluons and Knots

• Authors: Alfredo Guevara
• arXiv: 2610.10673
• Cross-list: hep-th
• Comments: 11 pages
• Subjects: High Energy Physics - Theory (hep-th); Geometric Topology (math.GT)

We argue that the scattering amplitude of a single minus helicity gluon with arbitrarily many positive helicity gluons is topological. The amplitude is constant in chambers of kinematic space, its resonances are located at walls and its factorization is determined by wall-crossing. Yangian invariance dictates the amplitude to be determined by a 2d null Wilson loop that can be lifted into a positive and transverse 3d link. By reinterpreting wall-crossing and the Weinberg soft theorem as skein relations and Reidemeister moves of the subknots, we construct a state-counting description consisting of adding the `corners' of their HOMFLY-PT polynomials. Beyond the classical limit, we argue that this provides a conjectured but precise connection to topological field and string theory.

The new formula is rigorously derived via a Lean-formalized dependency graph (DAG) as detailed in a companion paper, laying out the precise assumptions including Reidemeister III invariance. It is further tested exhaustively, including novel `star' kinematics where the amplitude and state count are shown to become Catalan numbers, in agreement with knot theory literature.

Two-generator presentations with a relation gap

• Authors: Laurent Bartholdi, Martin R. Bridson, Roman Mikhailov, John Nicholson
• arXiv: 2610.23637
• Cross-list: math.GR
• Comments: 3 pages
• Subjects: Group Theory (math.GR); Algebraic Topology (math.AT); Geometric Topology (math.GT)

We exhibit two-generator two-relator group presentations that have cyclic relation modules but do not present one-relator groups; this resolves the relation gap problem. By attaching a single 3-cell to each of the corresponding presentation 2-complexes, one obtains counterexamples to Wall's D2 problem.