math.GT daily digest: 17 submissions for 8 October 2026

New submissions

Surface Braid Invariants from Yang-Baxter Operators and Their Cohomology

• Authors: Masahico Saito, Emanuele Zappala
• arXiv: 2610.08837
• Comments: 30 pages, 16 figures. Comments are welcome!
• Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA)

Surface braids are a generalization of braids to surfaces in the 4-disk. They are described by planar graphs called charts, and sequences of braid words called braid systems, that represent monodromies of branch points in projections. Yang-Baxter operators (YBOs) have been used extensively in knot theory for constructions of Jones and other polynomials, defined by braid group representations. We propose to use YBOs for constructing surface braid invariants through braid systems. The intersections of eigenspaces are used for this construction. We further use 3-cocycles of the Yang-Baxter cohomology theories via charts, in an analogue of quandle cocycle invariants but more generally utilizing tensor products of modules. An approach towards defining knotted surface invariants through closures of surface braids is discussed.

Asymptotic enumeration of minimally intersecting filling curve systems on closed surfaces

• Authors: Sayantika Mondal
• arXiv: 2610.09102
• Comments: 23 pages, 1 figure. Code and data are provided as ancillary files
• Subjects: Geometric Topology (math.GT); Combinatorics (math.CO)

Let be the closed oriented surface of genus , and let $\Gamma$ be a finite collection of closed curves on that fills, in the sense that its complement is a union of disks. The total number of double points of such a $\Gamma$ in minimal position is at least ; we call $\Gamma$ minimally intersecting when equality holds. We prove that the number of mapping class group orbits of minimally intersecting filling curve systems satisfies as . We also study the single-curve subproblem, in which $\Gamma$ consists of a single component. Every such curve is in particular a filling system, so the corresponding count satisfies , which improves the upper bound previously known for single filling curves. Finally, we compute and exactly for small genus and conjecture an asymptotic growth rate.

Cabling formula and colored knot Floer homology

• Authors: Chen Zhang
• arXiv: 2610.09201
• Comments: 40 pages. Comments welcome!
• Subjects: Geometric Topology (math.GT)

We study colored knot Floer homology via a bordered cabling construction. We construct a cabling module and identify the connecting maps in the defining direct system at the level of this module. By taking the colimit at the chain level, we obtain a general formula for colored knot Floer homology of an arbitrary knot in terms of its ordinary knot Floer homology and the colored unknot algebra. This extends the computation of Alishahi-Gorsky-Liu from L-space knots to all knots.

On unifying Jones representations of braid groups and Thompson's group

• Authors: Yangxiao Luo, Shunyu Wan
• arXiv: 2610.09333
• Comments: 32 pages, 18 figures
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR); Quantum Algebra (math.QA)

We unify Jones' representations of braid groups and the oriented Thompson's group by introducing the oriented dyadic braid group. This finitely generated group contains every braid group and the oriented Thompson's group. We construct an embedding of the oriented Thompson's group into this group such that taking braid closures recovers Jones' construction of oriented links. We also introduce the dyadic Temperley-Lieb algebra and construct a unitary representation of the oriented dyadic braid group extending the classical Jones representations of braid groups. Along the Thompson group's embedding, its distinguished matrix coefficient agrees with the corresponding coefficient of Jones' representation of the oriented Thompson's group.

The top coefficient of the Links-Gould invariant is Murasugi-sum multiplicative

• Authors: Cristina Anghel, Daniel López Neumann, Roland van der Veen
• arXiv: 2610.09354
• Comments: 17 pages
• Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA)

We prove that the Laurent polynomial in that is the top coefficient of the Links-Gould invariant of the boundary of a Seifert surface is multiplicative under arbitrary Murasugi sums of minimal genus Seifert surfaces. This extends a previous result of the second and third authors which only holds for plumbing, i.e. 4-Murasugi sum. Our technique relies on quantum supergroup themselves and PBW basis rather than the skein relations used in the previous work.

-folded surfaces and the -Thurston norm

• Authors: William Hobkirk
• arXiv: 2610.09422
• Comments: 102 pages, 30 figures
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT)

-folded surfaces are a class of oriented folded surfaces with a fundamental class. In this paper we study the -folded surfaces that embed in $3$-manifolds. This perspective naturally leads to the definition of the -Thurston norm, which generalises the -Thurston norm of Jaco, Rubinstein and Tillmann and adapts Turaev's definition of $\theta$. We define canonical -folded surfaces, which connect the theory of -folded surfaces to the study of one-vertex $3$-manifold triangulations, and we relate their Euler characteristic to the -Thurston norm. We generalise a classical result of Bredon and Wood by proving that the maximum Euler characteristic of a non-separating non-empty -folded surface embedded in $L(3n,1)$ is for , determining the -Thurston norm in these cases. We conclude by demonstrating the motivating application of this theory to the study of generalised triangulations by proving that any generalised triangulation of the lens space $L(3n,1)$ must have at least $n$ tetrahedra. This final result is a preview of joint work with Spreer in which we determine the exact complexity of $L(3n,1)$ using more sophisticated techniques.

Fixed-point sets of semifree group actions on infinite-dimensional compact convex sets and their manifolds

• Authors: Jan van Mill, James E. West
• arXiv: 2610.09561
• Subjects: Geometric Topology (math.GT)

Let $A$ be the fixed point set of a semifree action of a compact (metric) group $G$ on a Keller space $X$ (an infinite-dimensional compact convex subset of a separable Fr' echet space). All Keller spaces are homeomorphic and all nontrivial compact Lie groups act on all $X$'s semifreely with unique fixed points (see Section 3). We prove (1) if $A$ is a single point, then it is the fixed point set of a semifree action of $G$ with uniformly small orbits, (2) if $A$ has Property Z (e.g., is of infinite codimension) in $X$, then the action of $G$ may be replaced by one with fixed point set $A$ and uniformly small orbits, and (3) if $A$ is an \ANR, then $G$ acts on $X$ semifreely with fixed point set homeomorphic to any compact (metric) space that is Shape equivalent to $A$. Corollaries are (a) every Cell-like set (i.e., continuum of trivial Shape) embedded in a Keller space $X$ as a Z-set is the fixed point set of semifree actions on $X$ with uniformly small orbits of all nontrivial compact Lie groups (Corollary \ref{CE}) and (b) (a partial converse to a Theorem of P.\ A.\ Smith) if a compact (metric) space $Y$ has the Shape of a compact \ANR, then $Y$ embeds in $X$ as the fixed point set of a homeomorphism of $X$ of period $n$ if it is acyclic in \v Cech homology with coefficients. Finally, we (4) apply recent results of Cappell, Weinberger, and Yan \cite{cwy} to give for a compact manifold $M$ modeled on a Keller space homological and algebraic K-theoretic conditions equivalent to the existence of semifree actions of finite groups $G$ on $M$ with prescribed \ANR\ fixed point sets and (5) generalize (3) to prove that if $A$ is an \ANR\ that is the fixed point set of a semifree action of $G$ on $M$, then every compact (metric) space that is Shape equivalent to $A$ is homeomorphic to the fixed point set of a semifree action of $G$ on $M$.

Purely cosmetic surgeries on knots with low-span Jones polynomials, II

• Authors: Kazuhiro Ichihara, Kengo Kishimoto
• arXiv: 2610.10019
• Comments: 10 pages
• Subjects: Geometric Topology (math.GT)

We show that a knot in the 3-sphere with a nontrivial Jones polynomial of span at most 14 admits no purely cosmetic surgery. This extends a result of the first author for span at most 11. We determine all Laurent polynomials of span 15 satisfying the conditions on the Jones polynomial used in the proof. These polynomials form an infinite family. We also show that a knot of braid index at most 4 whose Jones polynomial has odd span admits no purely cosmetic surgery.

Finite Chains of Two-Complexes and Acyclic Covers

• Authors: Laurent Bartholdi, Roman Mikhailov
• arXiv: 2610.10074
• Comments: Produced with help from AI
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

A negative answer to Whitehead's asphericity problem would follow from an infinite chain of two-complexes, starting with a non-aspherical one, in which every inclusion induces the zero map on second homotopy groups. We prove that a connected two-complex $K$ is the first term of such chains of every \emph{finite} length if and only if $K$ has a connected acyclic regular cover. In particular, there is a finite non-aspherical two-complex $K$, a presentation complex of $\mathrm{SL}(2,5)$, such that for every $n$ there is a strictly increasing chain of finite two-complexes in which all inclusions are zero on .

Computations of the slice genus and the unknotting number of links via machine learning

• Authors: Yutong Dai, Oliver Hayman, András Juhász, Ludovico Morellato
• arXiv: 2610.10206
• Comments: 72 pages, 34 figures
• Subjects: Geometric Topology (math.GT); Machine Learning (cs.LG); Machine Learning (stat.ML)

Links are disjoint unions of circles smoothly embedded in . We use reinforcement learning and Bayesian optimisation to obtain new upper bounds on several link invariants that are not known to be algorithmically computable: the slice genus and the unknotting number for links, and the strong slice genus for algebraically split links. We also compute lower bounds using known invariants. Combining the upper and lower bounds, we obtain new exact values in many cases. Our unknotting agents can reproduce the non-additivity of the unknotting number for several counterexamples due to Brittenham and Hermiller, in some cases finding new unknotting trajectories.

On the -extension of Bar-Natan multicurves

• Authors: Tomás Mejía-Gómez
• arXiv: 2610.10262
• Comments: 30 pages, 4 figures
• Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA)

Kotelskiy, Watson and Zibrowius show that the Bar-Natan invariant \unicode{x0414}(T)! associated to a $4$-ended oriented tangle $T$ extends to a twisted complex \unicode{x0414}^\infty(T)! over an -enhancement of the Bar-Natan algebra. We prove that \unicode{x0414}^\infty(T)! itself is a tangle invariant, settling a conjecture from their work. We deduce this from an invariance statement for a \emph{matrix multifactorization} $\mathcal{M}(T)$, generalizing a construction of Ballinger to the coefficient ring $\mathbb{Z}[G]$. This multifactorization is well defined up to a notion of 1-homotopy equivalence. The proof rests on explicit formulas for special deformation retracts of Koszul matrix factorizations, which realize delooping in this setting and may be of independent interest.

A Semiclassical Limit of Large-Colour State Sums for Semisimple Lie Algebras

• Authors: Atsuhide Nagasaka
• arXiv: 2610.10269
• Subjects: Geometric Topology (math.GT)

The Melvin--Morton--Rozansky theorem relates the large-colour semiclassical limit of the coloured Jones polynomial to the inverse Alexander polynomial. Bar-Natan and Garoufalidis [BNG96] proved the theorem using weight systems and, more generally, established the corresponding formula for arbitrary complex semisimple Lie algebras.

We give a direct proof of this semisimple Lie algebra generalization by analysing a state sum. For a complex semisimple Lie algebra $\mathfrak{g}$ and a dominant integral weight $\lambda$, we use the embedding to analyse the state sum. Under the specialization , we show that, as $d\to \infty$, only the identity terms and those proportional to contribute at leading order. It follows that the leading-order contribution decomposes over the positive roots, with each root contribution evaluated as in the case. Writing , the limit is therefore given by the product of over the positive roots $\alpha$.

Zero-surgery characterises Whitehead doubles

• Authors: Alice Merz, Diego Santoro
• arXiv: 2610.10338
• Comments: 17 pages, 6 figures. Comments are welcome!
• Subjects: Geometric Topology (math.GT)

We show that zero is a characterising slope for the untwisted Whitehead double of any knot $K$. More generally, this holds for all $n$-twisted Whitehead doubles of $K$, with the assumption that if $K$ is a non-trivial $(p,q)$-torus knot or $(p,q)$-cable. In particular, zero is a characterising slope for all twist knots.

Cross submissions

Quadratic differentials in Teichmüller theory and complex harmonic maps

• Authors: Christian El Emam, Nathaniel Sagman
• arXiv: 2610.08885
• Cross-list: math.DG
• Subjects: Differential Geometry (math.DG); Geometric Topology (math.GT)

We show that Hitchin-Wolf's harmonic map parametrization of Teichmüller space, the Schwarzian holonomy map, and Uhlenbeck's almost-Fuchsian holonomy map all arise as restrictions of a single locally biholomorphic map into the $\mathrm{PSL}(2,\mathbb{C})$ character variety. The map admits a right inverse defined on an open subset of quasi-Fuchsian space, encoding harmonic maps together with the Bers embedding. The construction goes through complex harmonic maps to the space of oriented geodesics of .

Along the way, we further develop the theory of complex harmonic maps, with a focus on the natural -action and the relation to complex projective structures. This leads to an independent application: we obtain a new geometric interpretation of certain real twistor lines in the rank-two Deligne-Hitchin moduli space and we recover a theorem of Gothen-Silva.

A continuum of coarse medians for all mapping class groups and right-angled groups

• Authors: Giorgio Mangioni
• arXiv: 2610.09162
• Cross-list: math.GR
• Comments: 23 pages, 2 figures. Comments are highly appreciated!
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT)

It is shown that the mapping class group of any non-sporadic surface admits a continuum of equivariant coarse median structures, thus improving a previous theorem of the author. The same holds for a wide class of right-angled Artin and Coxeter groups, including examples of any dimension at least two, and RACGs which are not quasi-isometric to RAAGs; the coarse median structures in these cases are not induced by cocompact cubulations. The results are instances of a general procedure, which uses quasicocycles to flexibly modify the structure of a (suitable) hierarchically hyperbolic group.

Derived Picard reconstruction for graded surfaces with stops

• Authors: Wen Chang, Dongjian Wu
• arXiv: 2610.09319
• Cross-list: math.RT
• Comments: 64 pages, comments welcome!
• Subjects: Representation Theory (math.RT); Geometric Topology (math.GT)

We study reconstruction of graded surfaces with stops from their mapping class groups and the derived Picard groups of their partially wrapped Fukaya categories. In genus at least two, we prove that the ungraded mapping class group determines the surface with stops $(S,M)$, while the derived Picard group with the shift distinguished determines the underlying graded surface $(S,\eta )$. Nevertheless, the derived Picard group with both the shift and the Serre functor distinguished has countably infinite fibres. We also construct arbitrarily large finite families of pairwise non-equivalent categories with the same invariant and Grothendieck rank.

For line fields induced by nowhere-vanishing vector fields, we show that the rational Serre representation determines the surface with stops $(S,M)$ in every genus. For arbitrary line fields in genus at least two, the full integral derived Picard representation, with the shift distinguished, has fibres of cardinality at most four. It completely determines the graded surface with stops when $S$ has one or at least five boundary components, while non-singleton fibres occur for two, three, and four boundary components.

A reduction theorem for Lê's conjecture

• Authors: Pablo Portilla Cuadrado
• arXiv: 2610.10418
• Cross-list: math.AG
• Comments: 24 pages, 7 figures
• Subjects: Algebraic Geometry (math.AG); Geometric Topology (math.GT)

Let be a holomorphic map germ admitting an injective representative. We prove that if , then . This reduces Lê's conjecture to excluding potential counterexamples $\mathbf{n}$ of orders $2$, $3$, and $4$. The proof combines techniques from the theory of plane curve singularities, explicit cobordism constructions, and genus bounds obtained by applying properties of the ${\rm Y}$ invariant coming from knot Floer homology.