math.GT daily digest: 5 submissions for 4 September 2026

New submissions

Quilt decompositions of surfaces and Torelli group action on the extended Hatcher complex

• Authors: Hiromasa Fuchizawa
• arXiv: 2609.02974
• Subjects: Geometric Topology (math.GT)
• Comments: 45 pages, 44 figures

We give a presentation of a groupoid of transformations of quilt decompositions of a surface, which are pants decompositions of the surface with the addition of "seams". We consider colored quilt decompositions of a surface, where one side of a quilt decomposition is colored. We also give a presentation of a groupoid of transformations of colored quilt decompositions of a surface. These presentations of such groupoids are realized by constructing connected and simply-connected two-dimensional cell complexes whose 0-cell are these up-to-isotopy structures on the surface. In the latter part of this paper, we show that Torelli group (or more precisely, a torsion-free subgroup of a mapping class group of a surface) acts on these complexes freely.

Fourier Analysis and Idempotents in Quandle Algebras

• Authors: Mohamed Elhamdadi; Luc Ta; Bryce Virgin
• arXiv: 2609.03799
• Subjects: Geometric Topology (math.GT); Functional Analysis (math.FA); Group Theory (math.GR); Rings and Algebras (math.RA)
• Comments: 13 pages

In 2023, the first author, Nunez, Singh, and Swain [10] formulated an analogue of Kaplansky's idempotent conjecture for integral quandle rings. In this paper, we develop a new Fourier-analytic approach to quandle rings and use it to establish the conjecture for two major classes of quandles: Takasaki quandles, including dihedral quandles, and medial commutative quandles. A central contribution of the paper is the introduction of Fourier analysis on Alexander quandles, which provides a new framework for studying quandle rings and, in particular, their idempotents. Using this Fourier-analytic framework, we also prove that a previously known sufficient condition for the existence of counterexamples to the conjecture is in fact necessary, thereby resolving a problem of Jablonowski [12].

Khovanov monodromy groups via motions

• Authors: Gabriel Corrigan; Livio Ferretti; Isacco Nonino; Susanna Terron
• arXiv: 2609.03932
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT); Group Theory (math.GR)
• Comments: 14 pages, 7 figures. Comments welcome!

For any link, we define a monodromy map from the motion group of the link to the group of automorphisms of the link's Khovanov homology. The image of this map is the \emph{unoriented monodromy group} of the link. This map allows us to convert results concerning motion groups of links into ones about their Khovanov monodromy. In particular, we use a characterisation of the motion groups of split links to write their unoriented monodromy groups as an explicit semidirect product in terms of their unsplit pieces. We give some demonstrative examples, computing the unoriented monodromy groups of unlinks, Hopf links, and split links composed of pieces thereof.

Cross submissions

On the Finiteness of Anosov flows on $3$-manifolds via Contact Geometry

• Authors: Jonathan Bowden; Vincent Colin
• arXiv: 2609.03700
• Cross-list from: Dynamical Systems (math.DS)
• Subjects: Dynamical Systems (math.DS); Geometric Topology (math.GT); Symplectic Geometry (math.SG)
• Comments: 26 pages, 6 figures

Following Eliashberg-Thurston and Mitsumatsu, one can associate a transverse pair of oppositely oriented contact structures to any Anosov flow. We show that the isotopy classes of these contact structures completely determine the flow up to isotopy orbit equivalence. This then implies that the number of Anosov flows modulo isotopy orbit equivalence on a closed hyperbolic $3$-manifold is finite. This approach also yields an explicit bound on the number of orbit equivalence classes of Anosov flows in terms of the number of universally tight contact structures. In addition, we show finiteness of pseudo-Anosov flows for which the complement of the singular orbits is atoroidal and, in the non-hyperbolic case, we obtain a control on the dynamics of Anosov flows on the hyperbolic pieces of the JSJ-decomposition.

Infinite chains of perfect fits for expanding Thurston maps

• Authors: Ino Loukidou
• arXiv: 2609.03838
• Cross-list from: Dynamical Systems (math.DS)
• Subjects: Dynamical Systems (math.DS); Complex Variables (math.CV); Geometric Topology (math.GT)
• Comments: 16 pages, 15 figures

The topological mating of two postcritically finite polynomials with dendritic Julia sets is encoded by a pair of circle laminations , whose collapse produces a sphere-filling curve. When a leaf of and a leaf of share an endpoint they form a perfect fit. An unpublished proposition of Epstein, recorded by Petersen and Meyer, shows that for matings of honest degree-$d$ polynomials an infinite-diameter ray equivalence class, i.e. an infinite chain of perfect fits is impossible. We show that this finiteness is a genuinely holomorphic phenomenon. Allowing the dynamics to carry a periodic critical orbit, we construct combinatorially expanding Thurston maps-realized by no expanding rational map-that admit invariant sphere-filling curves yet whose laminations contain infinite chains of perfect fits, in fact infinitely many of them.