math.GT daily digest: 12 submissions for 10 September 2026

New submissions

The real quotient of the Cartwright-Steger surface

• Authors: András I. Stipsicz; Zoltán Szabó
• arXiv: 2609.09207
• Comments: 13 pages
• Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG)

Borisov and Yeung showed that the Cartwright-Steger surface $X$ is defined over $\mathbb{Q}$, hence complex conjugation gives an antiholomorphic involution on $X$. We show that the orbit space $Y$ is a closed oriented smooth four-manifold homeomorphic to , presenting $X$ as a double branched cover , with branch locus diffeomorphic to the connected sum of three copies of the real projective plane.

Contractibility of the complex of incompressible Seifert surfaces: the knot case of Kakimizu's problem

• Authors: Guancheng Pan; Chengsong You; Junwei Zhou; Yongchao Chen
• arXiv: 2609.09224
• Comments: 55 pages
• Subjects: Geometric Topology (math.GT)

Let be a non-trivial knot and let $IS(K)$ be the simplicial complex whose vertices are the ambient isotopy classes of incompressible Seifert surfaces in the exterior $E(K)$, a finite set of distinct vertices spanning a simplex exactly when its classes admit simultaneously pairwise disjoint representatives. Kakimizu proved that $IS(K)$ is connected; whether it is contractible was asked by Przytycki and Schultens, who also identified the obstruction, namely that the projection used in the connectedness proof is not known to be well defined on isotopy classes. We prove that $IS(K)$ is contractible, and likewise every truncation on the vertices of genus at most $\ell$. The argument factors through a purely combinatorial theorem: every non-empty connected flag exchange complex is contractible, where an exchange complex carries a complexity function subject to two axioms which require only that an exchanging vertex exist, never that one be selected. That is what circumvents the obstruction. We locate the combinatorial theorem precisely against the existing literature -- it is not implied by dismantlability, and, when all descending links are finite, it follows from Zaremsky's descending-link criterion by a reindexing (Section 7.4) -- and we record what remains genuinely open. Two extensions are proved: the truncations above, and the case of links satisfying a linking condition that forces every spanning surface to be connected.

Computation of unknotting numbers: which knot breaks the Bernhard-Jablan Conjecture

• Authors: Seong-Jin Lee
• arXiv: 2609.09861
• Subjects: Geometric Topology (math.GT)

We determine $2,525$ unknotting numbers of prime knots with at most $13$ crossings, that are unknown in the KnotInfo snapshot of 9 September 2026. The lower-bound calculations use Heegaard Floer correction-term obstructions and Greene's spanning-tree model. We also give explicit crossing-change constructions for upper bounds. For $959$ alternating knots with range $[2,3]$, we verify that no crossing change in a minimal diagram gives a knot of unknotting number one. We determine the unknotting numbers of all four knots in Brittenham and Hermiller's construction and thereby identify 13n3370 as an explicit counterexample to the original Bernhard-Jablan conjecture.

Links and singularities of stable maps from $3$-manifolds to surfaces

• Authors: Gakuto Kato
• arXiv: 2609.09937
• Comments: 19 pages, 21 figures
• Subjects: Geometric Topology (math.GT)

For any two links in the $3$-sphere, we provide a visual construction of a stable map $f$ from the $3$-sphere to the Euclidean plane such that $f$ has no cusp points, the set of definite (resp. indefinite) fold points of $f$ is isotopic to the first (resp. second) given link, and $f$ does not have a certain type of singular fibers. Moreover, we obtain a similar result for any $3$-manifold by setting the target space to the $2$-sphere.

Hopf decomposition of the actions of subgroups of the mapping class group

• Authors: Hideki Miyachi
• arXiv: 2609.10175
• Comments: 38 pages
• Subjects: Geometric Topology (math.GT); Complex Variables (math.CV); Dynamical Systems (math.DS)

We study the Hopf decomposition of subgroup actions of the Teichmüller modular group on the Thurston boundary with respect to the Thurston measure class. Kaimanovich's general Radon--Nikodym criteria allow us to describe the conservative and dissipative parts by the divergence and convergence of a series expressed in terms of extremal length. We identify the conservative part with the big horospherical limit set modulo null sets. For any basepoint with trivial stabilizer in the subgroup, we identify the dissipative part, modulo null sets, with the set of Dirichlet points and with the union of the subgroup translates of the ideal boundary of the associated Dirichlet polyhedron. The description via Dirichlet polyhedra uses the fact that level sets of extremal-length ratios at distinct points of Teichmüller space have measure zero.

For the Torelli group of a closed surface of genus at least two, we use radial limits of the period map to prove that its conical limit set has measure zero. Combining our geometric characterization with the conservativity of its boundary action established by Choi, Gekhtman, Yang, and Zheng, we obtain that its big horospherical limit set has full measure and that the ideal boundary of each Dirichlet polyhedron has measure zero.

Extremal entropy for products of Fuchsian representations

• Authors: Richard Canary; Dongryul Kim
• arXiv: 2609.10325
• Subjects: Geometric Topology (math.GT); Dynamical Systems (math.DS)

In this paper, we count the number of (almost) extremally stretched closed geodesics for a pair of (non-conjugate) Fuchsian representations of a closed surface group, and show that it has subexponential growth. We then deduce that, as a discrete subgroup of , the growth indicator of the product representation vanishes on the boundary of the Benoist limit cone. We also prove that for a general Zariski dense Borel Anosov subgroup of , its growth indicator vanishes on at least one boundary component of the Benoist limit cone, but not necessarily on both. One may view our first result as a sharpening of Thurston's result that there is a unique geodesic lamination $\lambda$ such that every measured lamination maximizing the ratio of lengths with respect to the two representations has support contained in $\lambda$. We hope this will be a starting point for a more general study of extremal entropy.

On Medial Quandle Coloring Link Invariant and Detecting Causality

• Authors: Hongxu Chen
• arXiv: 2609.10449
• Subjects: Geometric Topology (math.GT)

We investigate the capability of medial quandles to detect causality in (2+1) dimensional globally hyperbolic spacetime by determining if their coloring link invariants can distinguish between the connected sum of two Hopf links and an infinite series of relevant three-component links constructed by Allen and Swenberg in 2020, who suggested that any link invariant must be able to distinguish those links for them to detect causality in the given setting. We show that these quandles fail to do so as long as defines an equivalence relation. The Alexander quandles are an example to which this result applies. We also give a sufficient condition for a medial quandle to distinguish between the connected sum of two Hopf links and all links in Allen-Swenberg series, and give an example of such a quandle. Inspired by this result, we also derive a generalized theorem about the coloring of medial quandles on a specific type of tangle, which helps to determine whether these quandles can distinguish between a wider range of knots and links or not.

Cross submissions

Determinantal representation of colored HOMFLY for double braids

• Authors: Ya. Kononov; A. Morozov
• arXiv: 2609.09437
• Cross-list: hep-th
• Comments: 18 pages
• Subjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Geometric Topology (math.GT)

Starting from the known differential expansions, we express the HOMFLY-PT polynomials of twist and antiparallel double-braid knots in rectangular representations as determinants of and matrices. We first give an elementary derivation for the figure-eight knot and then re-sum the KNTZ formulas for twist and double-braid evolution. The resulting matrices are constructed from single-hook evolution coefficients and explicit representation-dependent weights. These formulas provide a starting point for investigating possible extensions of knot polynomials to KP/Toda $\tau$-functions. Extensions to non-rectangular representations and to more general knot families remain open problems.

The hyperbolic class and vanishing laws in a TMF-valued four-manifold invariant

• Authors: Yuqi Li; Hao-Yu Sun
• arXiv: 2609.09819
• Cross-list: math.AT
• Comments: 15 pages
• Subjects: Algebraic Topology (math.AT); High Energy Physics - Theory (hep-th); Geometric Topology (math.GT)

Under explicit hypotheses on the spectral Looijenga construction, we compute the hyperbolic-plane value of the zero-section invariant of Gukov, Krushkal, Meier, and Pei in periodic topological modular forms. The value is the Hopf element eta, and adjoining a hyperbolic plane acts by multiplication by this element. Earlier work obtains the hyperbolic value under an additional cobordism-duality assumption; here we derive it directly from the Looijenga restriction maps without that assumption. Together with annihilation and vanishing results for definite lattices, this gives a complete value formula on smooth closed simply connected spin four-manifolds. In particular, nonzero signature forces the invariant to vanish, so both orientations of a K3 surface have value zero.

Overlap-Helly theorems

• Authors: Andreas F. Holmsen; Alfredo Hubard
• arXiv: 2609.10023
• Cross-list: math.CO
• Comments: 17 pages
• Subjects: Combinatorics (math.CO); Computational Geometry (cs.CG); Geometric Topology (math.GT)

In this paper we introduce a generalization of Helly's theorem closely connected to Bárány-Gromov overlap theorems (also called selection lemmas). Our main result implies both the topological colorful Helly of Kalai and Meschulam and Karasev's topological centerpoint theorem. We further investigate the topological fractional Helly theorem from this overlap perspective, and show an overlap theorem for dense complexes (a continuous second selection lemma for tame maps).

Superpolynomial lower bounds for vertex numbers of real projective space triangulations via a topological Figiel-Lindenstrauss-Milman theorem

• Authors: Florian Frick; Kaave Hosseini; Eric Myzelev; Arya Narnapatti; Aliaksei Vasileuski
• arXiv: 2609.10402
• Cross-list: math.CO
• Comments: 16 pages
• Subjects: Combinatorics (math.CO); Geometric Topology (math.GT)

We prove that every simplicial triangulation of real projective $d$-space has vertices. Together with known constructions, this determines the minimum vertex number as . The result follows from a topological generalization of the Figiel--Lindenstrauss--Milman inequality, answering a recent question of Frick, Hosseini, and Vasileuski: a finite strongly regular CW complex with a free cellular involution, $v$ vertices, and $f$ maximal cells has $\mathbb{Z}/2$-index at most . We bound the dimensions of Morse cells by a trace estimate for a constrained Hessian, obtaining a Morse-theoretic proof of the classical inequality for centrally symmetric polytopes. As further applications of this inequality, we give an lower bound for the order of a triangle-free topologically $t$-chromatic graph and bound the index of sign complexes by for total matrices and for partial matrices, where is the number of columns and is the VC dimension.

Small undecidable groups and unrecognizable 4-manifolds

• Authors: Marc Kegel; Shana Yunsheng Li; Qiuyu Ren
• arXiv: 2609.10461
• Cross-list: math.GR
• Comments: 14 pages; the main results were developed by ChatGPT 5.6 Sol
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT)

We construct a $3$-generator $9$-relator group with unsolvable word problem. We use the group to construct two fixed-size Adian--Rabin families of group presentations, one with $4$ generators and $11$ relators, and another with $2$ generators and $10$ relators. As a consequence, is topologically unrecognizable and is smoothly unrecognizable. These algebraic and topological results improve the previous best known bounds by Borisov, Tancer, and Gordon.

The construction of the group builds upon an example of Borisov and uses additional HNN extensions and Tietze eliminations to reduce the size of the presentation. We also provide a machine-checked Lean 4 formalization of the algebraic results.