math.GT daily digest: 6 new submissions for 14 August 2026

New submissions

Torsion in skein modules and shared knot surgeries

• Authors: Efstratia Kalfagianni
• arXiv: 2608.12668
• Subjects: Geometric Topology (math.GT)

For every , we construct a closed Haken 3-manifold $M\_g$, whose only connected, incompressible surface is a non-separating surface of genus $g$ and discuss the torsion behavior of the Kauffman bracket skein module §(M\_g,\\Z\[A^{\\pm 1}\])!. We also use properties of shared surgeries between knots to estimate (and often compute) the dimension of the Kauffman bracket skein module over $Q(A)$, for families of hyperbolic 3-manifolds obtained by surgery on double twist knots. We have similar computations for some surgeries on several knot complements in the SnapPy census of cusped hyperbolic manifolds with triangulations up to 9 tetrahedra.

On the minimal diameter of finite covers

• Authors: Yifei Cai; Qiliang Luo
• arXiv: 2608.12887
• Subjects: Geometric Topology (math.GT)
• Comments: 19 pages, 2 figures

We prove that every closed hyperbolic surface admits a sequence of finite covers with asymptotically optimal diameters.

Reconstructing Turaev--Viro Invariants from Four-Fold Simple Branched Covering Presentations

• Authors: Eri Hatakenaka
• arXiv: 2608.13241
• Subjects: Geometric Topology (math.GT)

We construct a finite state sum from a link diagram labelled by transpositions. The diagram presents a closed 3-manifold as a four-fold simple branched cover, and the state sum equals the SU(2)-type Turaev--Viro invariant of that manifold. For each admissible colouring of the diagram's edges and complementary regions, a global factor is combined with crossing weights obtained by summing over compatible local colourings. We then sum these products over all diagram colourings. The formula is proved by assembling crossing-wise local triangulations and matching the two state sums.

Limit sets of mapping class groups

• Authors: Hideki Miyachi; Ken'ichi Ohshika
• arXiv: 2608.13354
• Subjects: Geometric Topology (math.GT)

We introduce the notions of horospherical limit point and big horospherical limit point in the projective measured foliation space for a subgroup of the mapping class group, which are analogous to these notions for hyperbolic spaces. We classify Teichmüller geodesic rays directed by projective measured foliations into conical, horospherical, big horospherical limit points, and show that these can be determined by the topological properties of the directing foliations: except for the case of minimal and uniquely ergodic projective measured foliation, whose corresponding ray can be either a conical limit point or a non-conical horospherical limit point.

Cross submissions

Monodromy of plane curve singularities and quiver mutation

• Cross-list: Algebraic Geometry (math.AG)
• Authors: Roger Casals
• arXiv: 2608.12484
• Subjects: Algebraic Geometry (math.AG); Combinatorics (math.CO); Geometric Topology (math.GT); Representation Theory (math.RT)
• Comments: 63 pages, 11 figures

The main result of this article shows that the quiver mutation class of a malleable real Morsification uniquely determines the integral monodromy module of a plane curve singularity. In particular, we show that the quiver mutation class of a malleable divide determines the complex topological type of an irreducible plane curve singularity. This establishes the algebraic-to-topological implication of a conjecture of S.~Fomin, P.~Pylyavskyy, E.~Shustin and D.~Thurston in the malleable irreducible case. The result is proven by developing representation-theoretic techniques based on an equivariant Euler pairing in the derived category of continuous finite-dimensional dg modules over a differential bigraded Ginzburg algebra. A key step uses these techniques to show that the quiver mutation class of a plabic fence uniquely recovers the torsion part of the Alexander module of the associated smooth link.

Laminations and External Angles for Similarity Pairs

• Cross-list: Dynamical Systems (math.DS)
• Authors: Danny Calegari; Alden Walker
• arXiv: 2608.12774
• Subjects: Dynamical Systems (math.DS); Geometric Topology (math.GT)
• Comments: 54 pages, 26 figures

A {\em similarity pair} is the dynamical system in $mathbbC$ generated by two maps and for . Associated to the dynamical system is an attractor $Lambda$. The Barnsley--Harrington Mandelbrot set is the set of for which $Lambda$ is connected. Let $K$ denote the filled set of a connected $Lambda$. For we show that the (partially defined) action of the semigroup on is topologically conjugate to a (discontinuous) piecewise linear action of constant slope. Conditional on a conjecture (satisfied for `most' ) we give a necessary and sufficient condition in terms of the dynamics on for $K$ to contain cut points, and we describe the set of all such cut points in terms of infinite walks in a directed graph obtained by an explicit recursive algorithm.

The structure of the `dynamical cut point set' for a 2-dimensional family of piecewise linear actions (containing those coming from ) recovers and generalizes the Douady--Hubbard--Thurston quadratic minor lamination for the abstract Mandelbrot set.