math.GT daily digest: 22 new submissions for 1 September 2026

New submissions

Fox $p$-Colorings as Fixed Points of Braid Representations

• Authors: Neungjoo Goh; Suah Jeong; Seungbin Oh; Hyungkee Yoo
• arXiv: 2608.29046
• Subjects: Geometric Topology (math.GT)

Fox $p$-colorings of knots and links admit a linear-algebraic description in terms of braid representations. For each braid we associate a representation such that Fox $p$-colorings of the closed braid correspond to fixed points of $M(\beta )$. As a consequence,

yielding an explicit formula for the number of Fox $p$-colorings of arbitrary knots and links. We apply the method to torus links and spiral links, obtaining explicit descriptions of their Fox 3-coloring spaces.

The crossing matrix and the Burau matrix of pure braids

• Authors: Ayaka Shimizu; Yoshiro Yaguchi
• arXiv: 2608.29147
• Subjects: Geometric Topology (math.GT)
• Comments: 11 pages, 2 figures

We show that the first derivative of the Burau matrix at $t=1$ coincides with the Laplacian of the crossing matrix for pure braids. We also discuss conjugacy invariants derived from these matrices.

Rigidity of the period map up to finite covers

• Authors: Xiyan Zhong
• arXiv: 2608.29351
• Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG)
• Comments: 54 pages, 8 figures. Comments welcome!

We first give a complete classification of bi-affine representations of mapping class groups of surfaces with finitely many boundary components or punctures. We also show that every linear representation of the mapping class group of a genus-$g$ surface with two boundary components of dimension at most is bi-affine.

We then classify low-dimensional symplectic representations of the mapping class group associated to triple covers. Let , and let be the corresponding triple cover with deck transformation $\sigma$. For , every non-abelian homomorphism from either , the stabilizer of $[\beta ]$ in , or , the centralizer of $\sigma$ in , to is, up to conjugation, the standard symplectic representation on .

As an application, we obtain a rigidity theorem for holomorphic maps from the moduli space of genus-$g$ curves equipped with a $3$-sheeted (unbranched) normal covering to the moduli space of $h$-dimensional principally polarized abelian varieties. We prove that, for and , the unique nonconstant holomorphic map from , equipped with either of its two natural complex-orbifold structures, to is the period map sending a cover to the Jacobian of the base curve $X$.

Cabled braids and their crossing matrices

• Authors: Ayaka Shimizu; Yoshiro Yaguchi
• arXiv: 2608.29486
• Subjects: Geometric Topology (math.GT)
• Comments: 15 pages, 12 figures

We study cabling operations on braids, and characterize the matrices that can be realized as crossing matrices of reducible pure braids. We also use cabling to construct enhanced conjugacy invariants of braids.

Cyclotomic Newton Expansions and a Rank-Uniform Integer-Valued Newton Completion

• Authors: Honghuai Fang; Tian Zhou
• arXiv: 2608.29585
• Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA); Representation Theory (math.RT)
• Comments: 48 pages, no figures

Let denote the reduced $SU(n)$ quantum invariant of a zero-framed knot $K$, colored by the $r$th symmetric power of the defining representation and normalized to be 1 for the unknot. For every fixed we prove the Chen--Liu--Zhu cyclotomic expansion conjecture: there are unique coefficients such that

where . The finite dual interpolation formula of Beliakova--Gorsky gives an integral one-sided factorial expansion. After identifying their reduced scalar with the Habiro--Lê convention, we restrict the completed center to one-row colors. Completed Harish--Chandra reflection then yields inversion symmetry in the variable $z$, and integral descent through converts the one-sided expansion into the two-sided Newton basis. Cyclotomic-local interpolation and a UFD denominator-removal argument prove Laurent integrality of the Newton coefficients.

We also determine a natural coefficient ring for a rank-uniform expansion. For every zero-framed knot there are unique Laurent differential coefficients . The associated Newton coefficients are Laurent polynomials in $A$ over $\mathbb{Q}(q)$ whose values at every geometric node , , lie in . They define a two-variable Newton inverse-limit element whose positive-rank specializations recover all symmetric-color HOMFLY--PT polynomials. The completion is taken in the Newton kernels rather than coefficientwise at roots of unity. After a positive rank and a color have been fixed, the series is finite and may be evaluated at a root of unity.

Reduced Khovanov-Rozansky homology

• Authors: David Popović
• arXiv: 2608.29747
• Subjects: Geometric Topology (math.GT)

We introduce a variant of reduced Khovanov-Rozansky homology defined in terms of the variables alpha_1, ..., alpha_{n-1} that correspond to the regions between the strands. We rigorously reestablish several folklore results in terms of our framework, including the equivalence between Khovanov's and Rasmussen's approaches. Furthermore, we show that Rasmussen's chain complex is a free resolution of the Rouquier chain complex.

Simple geodesics in closed hyperbolic manifolds

• Authors: Qiliang Luo; Vladimir Marković
• arXiv: 2608.29761
• Subjects: Geometric Topology (math.GT)

Given any closed hyperbolic $n$-manifold $M$ (), we show that for a fixed , a random closed geodesic of length close to $r$ has the collar of width when $r$ is large enough. In particular, most closed geodesics of length close to $r$ are simple. This theorem positively answers Problem 3.16 from Kirby's list which asks whether every closed hyperbolic 3-manifold contains infinitely many simple closed geodesics.

Growth of Simple Closed Geodesics in closed Hyperbolic $3$-Manifolds

• Authors: Xiaolong Hans Han; Bohan Yang
• arXiv: 2608.29855
• Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG); Dynamical Systems (math.DS)
• Comments: 23 pages

Let $M$ be a closed hyperbolic $3$-manifold. We prove that the number of primitive simple closed geodesics of length at most $L$ is as $L\to \infty$.

Khovanov-Rozansky homology over

• Authors: David Popović
• arXiv: 2608.29986
• Subjects: Geometric Topology (math.GT)

We define a 'full' version of Khovanov-Rozansky homology: a chain complex CH(K) over F[U,V] whose chain homotopy type is a knot invariant and which has the property that setting U = V = 0 recovers the reduced Khovanov-Rozansky homology. We explore the algebraic structure of CH(K) and show that a variation of the structure theorem for knot Floer homology applies. This allows us to define counterparts to knot Floer concordance invariants $\tau$, $ϵ$ and in the Khovanov-Rozansky setting.

Ziggurats, taut foliations, and contact structures

• Authors: Thomas Massoni; Jonathan Zung
• arXiv: 2608.30031
• Subjects: Geometric Topology (math.GT); Dynamical Systems (math.DS); Symplectic Geometry (math.SG)
• Comments: 130 pages, 48 figures. Comments welcome!

We study the geography of taut foliations on compact $3$-manifolds with toroidal boundary components. The central object of our work is the set of boundary multislopes realized by foliations transverse to a fixed flow on such a manifold. We prove that these sets exhibit remarkable structural properties (rationality, rigidity, and convexity) which motivate the name ziggurats. Our main tool, of independent interest, is a two-way correspondence between foliations and contact structures on $3$-manifolds with boundary: we generalize the Eliashberg-Thurston theorem, which produces pairs of positive and negative contact structures from foliations, and a construction of the first author, which builds foliations from such contact pairs. Using both directions of this correspondence, we bring contact-geometric methods to bear on the architecture of ziggurats.

R-equivalence of quandle colorings and inner automorphisms

• Authors: Mai Sato
• arXiv: 2608.30132
• Subjects: Geometric Topology (math.GT)
• Comments: 9 pages, 5 figures

R-equivalence is an equivalence relation on the set of colorings of an oriented knot diagram by a quandle. In this paper, we show that a certain subgroup of the inner automorphism group of a quandle acts on the R-equivalence class of a given coloring by the quandle. We also determine the R-equivalence classes of colorings of a diagram of a $2$-bridge knot by a dihedral quandle completely, under a certain condition.

Brieskorn spheres bound orbit spaces of torus manifolds

• Authors: Grigory Solomadin
• arXiv: 2608.30513
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT)
• Comments: 11 pages, 4 figures; comments are welcome

We construct a smooth, equivariantly formal, simply connected torus $8$-manifold over a contractible $4$-manifold with the boundary a Brieskorn homology $3$-sphere having a nontrivial fundamental group. We prove extension to a torus graph for GKM graphs of GKM manifolds in complexity $2$, and $3$ under a finite fundamental group assumption (that is not satisfied, in general).

On the Nielsen realisation problem for cyclic groups of prime order

• Authors: Christian Kremer
• arXiv: 2608.30620
• Subjects: Geometric Topology (math.GT)
• Comments: 35 pages, comments welcome

We solve the Nielsen realisation problem for high-dimensional aspherical manifolds in a new class of special cases. Mainly, we focus on actions of cyclic groups of prime order. A novelty is that it gives topological solutions to the problem with certain high-dimensional fixed point sets, whereas previous solutions of this type were restricted to discrete fixed point sets. This is enabled by recent joint work with Kirstein on isovariant Poincaré duality spaces, and Farrell-Lück-Steimle's obstruction theory to finding approximate fibrations within a homotopy class.

L-space surgeries on (1,1)-knots in

• Authors: Qingfeng Lyu; Zipei Nie
• arXiv: 2608.30925
• Subjects: Geometric Topology (math.GT)
• Comments: 21 pages, 10 figures

We extend the diagrammatic criterion for $(1,1)$ L-space knots by Greene, Lewallen, and Vafaee to $(1,1)$-knots in . We also discuss applications to L-space surgeries on two-bridge links.

Pseudo-Anosov flow and dynamics on guts

• Authors: Yu Huang
• arXiv: 2608.30973
• Subjects: Geometric Topology (math.GT); Dynamical Systems (math.DS)
• Comments: 50 pages, 20 figures. Comments are welcome!

We relate the dynamics on a closed 3-manifold to the topological invariant, homology guts, proposed by Agol and Zhang. Given a closed manifold $M$ that admits a pseudo-Anosov flow $\phi$ without perfect fits, we construct a canonical semiflow on the homology guts associated to homology classes carried by $\phi$. We show that, up to orbit equivalent outside half annuli, the semiflow on the guts is invariant on each Thurston open cone. As an application, the fundamental group and the sutured structure of homology guts encode closed orbits of $\phi$ with particular type. Besides, for a positive class in , the canonical semiflow on the guts has a well-defined growth rate.

Cross submissions

Symplectic Tiling Billiards on Complete Affine Tori

• Authors: Charles Daly; Fabian Lander
• arXiv: 2608.28894
• Cross-list from: Dynamical Systems (math.DS)
• Subjects: Dynamical Systems (math.DS); Geometric Topology (math.GT); Symplectic Geometry (math.SG)
• Comments: 46 pages, 30 figures

In 2023, Richard Schwartz introduced a new dynamical system which is a marriage of two types of familiar billiards, tiling billiards and symplectic billiards. In this paper we investigate this dynamical system played on tilings of the plane which arise from non-Euclidean geometries on the torus. We review the affine analogue of the flat conformal structures on the torus through the work of Oliver Baues and William Goldman, and define an open subset of this deformation space corresponding to markings of complete affine tilings of the plane. We make this definition precise, and provide algebraic conditions on the symmetries of the tiling to define it. We then analyze the dynamics of symplectic tiling billiards played on these types of tilings and investigate the long-term dynamics of the system to prove a stability result concerning divergent trajectories. The divergence is defined in terms of geometric invariants arising from the tiling symmetry group. We argue that in some sense this divergence is a consequence of the tiles of a non-Euclidean tiling becoming 'thin' as one moves far away in the tiling. To do so we introduce a notion of thinness that is well adapted to the non-Euclidean affine tilings.

Weighted Homology and Cohomology of Weighted Polyhedra

• Authors: Yin Wei; Lisu Wu; Li Yu
• arXiv: 2608.29013
• Cross-list from: Algebraic Topology (math.AT)
• Subjects: Algebraic Topology (math.AT); Geometric Topology (math.GT)
• Comments: 46 pages, 7 figures, some contents overlap with arXiv:2106.06794

We define the notion of weighted polyhedron which can be thought of as the geometric realization of a weighted simplicial complex introduced by Dawson. Moreover, we will define a weighted version of singular homology theory for a weighted polyhedron and prove that it is isomorphic to the weighted simplicial homology of the weighted polyhedron. This implies that weighted simplicial homology is an invariant under isomorphisms and more generally under certain type of homotopy equivalences of weighted polyhedra. Moreover, we will generalize the cup product and cap product to weighted singular cohomology. In addition, we will interpret some known theories of orbifolds in terms of our weighted singular homology and cohomology.

Triangulated polygons and Y-frieze patterns

• Authors: Hin Chung Henry Tsang; Jon Wilson
• arXiv: 2608.29655
• Cross-list from: Combinatorics (math.CO)
• Subjects: Combinatorics (math.CO); Geometric Topology (math.GT); Rings and Algebras (math.RA)
• Comments: 30 pages, 2 figures

In the spirit of Conway and Coxeter, we classify all $\mathbf{Y}$-frieze patterns of type . In particular, we settle a conjecture made by de Saint Germain that all such $\mathbf{Y}$-frieze patterns arise from Conway-Coxeter frieze patterns. Moreover, our approach naturally leads to the enumeration of these $\mathbf{Y}$-frieze patterns in terms of Fuss-Catalan numbers.

Taut foliations through a contact lens

• Authors: Thomas Massoni
• arXiv: 2608.29989
• Cross-list from: Symplectic Geometry (math.SG)
• Subjects: Symplectic Geometry (math.SG); Dynamical Systems (math.DS); Geometric Topology (math.GT)
• Comments: 22 pages, 1 figure. Written for the proceedings of the 2025 Georgia International Topology Conference. Comments welcome!

This survey explores the rich interplay between (taut) foliations and (tight) contact structures in dimension three, highlighting recent work of the author. We outline a new method for constructing taut foliations from suitable pairs of contact structures, and discuss some applications and future research directions. In particular, we give a brief account of work in progress with Jonathan Zung on ziggurats for taut foliations transverse to pseudo-Anosov flows, and we propose a (mostly speculative) contact perspective on the $L$-space conjecture.

Compactification of $\mathrm{SL}(3,\mathbb{C})$-Character Varieties of Surfaces via Skein Algebras

• Authors: Seong Youn Kim
• arXiv: 2608.30217
• Cross-list from: Algebraic Geometry (math.AG)
• Subjects: Algebraic Geometry (math.AG); Combinatorics (math.CO); Geometric Topology (math.GT)
• Comments: 52 pages, 14 figures

We investigate the filtration structure of the skein algebra induced by the (quantum) trace map. Utilizing the structure, we prove that relative $\mathrm{SL}(3,\mathbb{C})$ character varieties of punctured surfaces admit log Calabi-Yau compactifications and prove the (weak) geometric P=W conjecture.

Quasi-convexity of energy functions along Teichmüller geodesics

• Authors: Inkang Kim; Xueyuan Wan; Genkai Zhang
• arXiv: 2608.30752
• Cross-list from: Differential Geometry (math.DG)
• Subjects: Differential Geometry (math.DG); Geometric Topology (math.GT)
• Comments: 57 pages. Comments are welcome

Hyperbolic length functions are among the most fundamental ones on Teichmüller space, and they are quasi-convex along Teichmüller geodesics. In this paper, we investigate the same question for energy functions of harmonic maps in two natural settings, which may be viewed as nonlinear and higher-dimensional analogs of the length functions. For a fixed domain and a varying hyperbolic target, we prove the energy is quasi-convex along Teichmüller geodesics under a filling hypothesis. Furthermore, we generalize Masur's result on asymptotic growth of the length function along the Teichmüller geodesic determined by a Jenkins-Strebel differential to the energy functions. We also prove the quasi-convexity for covering maps between closed hyperbolic surfaces with fixed target and varying domains. We derive first and second variation formulas of energy functions along Teichmüller geodesics and explain why the natural global statement is quasi-convexity rather than genuine convexity.

Fundamental groups of asymptotic cones of Lie groups with the SOL obstruction

• Authors: Antoine Velut
• arXiv: 2608.30767
• Cross-list from: Group Theory (math.GR)
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT); Metric Geometry (math.MG)
• Comments: 43 pages, 8 figures

We study asymptotic cones of Lie groups, presenting a link between the geometry of the weights and the highly non-simply-connected nature of asymptotic cones. We show that a Lie group that admits a group of SOL-type as a quotient is such that the fundamental group of its asymptotic cone contains the fundamental group of the Hawaiian earring space. We also obtain a strong non-vanishing result for the first homology group of the asymptotic cone. The assumption on the Lie group, introduced by Abels and called the "SOL obstruction", is equivalent to an explicit geometric condition on the weights. Our work builds upon the results of Burillo, who proves the statement on asymptotic cones in the case of SOL, and on those of Cornulier and Tessera who show that the SOL obstruction implies exponential growth of the Dehn function.