math.GT daily digest: 20 submissions for 22 September 2026

New submissions

The Conway knot has infinite order in smooth concordance

• Authors: Marco Golla; Juanita Pinzón-Caicedo
• arXiv: 2609.22378
• Subjects: Geometric Topology (math.GT)

We prove that the Conway knot has infinite order in the concordance group, and in particular that it is not slice. Ours is an alternative, independent proof with respect to those of Piccirillo and of Donatone, Kegel, Lewark, and Truöl.

The proper geometric dimension of the mapping class groups of non-orientable surfaces with punctures

• Authors: Nestor Colin; Rita Jiménez Rolland; Porfirio L. León Álvarez; Luis Jorge Sánchez Saldaña
• arXiv: 2609.22505
• Comments: 19 pages
• Subjects: Geometric Topology (math.GT)

We study the proper geometric dimension of {\it full} mapping class groups of non-orientable surfaces with punctures. Building on the computation for closed non-orientable surfaces, we prove that the proper geometric dimension agrees with the virtual cohomological dimension in all but a finite collection of low-complexity cases, for which we obtain explicit bounds. Our results provide the non-orientable counterpart of the corresponding computations for mapping class groups of closed and punctured orientable surfaces.

Cut pairs and Morse splitting of finitely generated groups

• Authors: Suzhen Han; Hao Liang; Qing Liu
• arXiv: 2609.22671
• Comments: 14 pages
• Subjects: Geometric Topology (math.GT)

Bowditch's theorem for hyperbolic groups establishes a fundamental correspondence between splittings over two-ended subgroups and the existence of local cut points in the Gromov boundary. While analogous results have been obtained for CAT(0) and relatively hyperbolic groups, no general theorem of this type exists for arbitrary finitely generated groups. The Morse boundary, introduced by Charney-Sultan and extended by Cordes, provides a quasi-isometry invariant boundary for any finitely generated group that naturally generalizes the Gromov boundary. In this paper, we prove that a splitting of a finitely generated group with connected Morse boundary over a two-ended Morse subgroup gives rise to a separating pair of points in the Morse boundary.

Simple Slice Surfaces

• Authors: Mark Pencovitch
• arXiv: 2609.22924
• Comments: 23 pages, 1 figure
• Subjects: Geometric Topology (math.GT)

Given a simply-connected $4$-manifold with boundary the $3$-sphere, we give sufficient conditions for a knot in the boundary to bound a simply embedded locally flat surface in the $4$-manifold of fixed nonzero genus, representing a fixed nonzero homology class. In certain indefinite cases, we also give conditions for these slice surfaces to be unique up to equivalence.

A polynomial invariant of planar curves

• Authors: Zhiyun Cheng; Yin Tian; Tianyu Yuan
• arXiv: 2609.23638
• Comments: 19 pages
• Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA)

We construct a polynomial invariant of planar curves via skein relations, which generalizes Arnold's numerical invariant $\mathrm{St}$.

Counting tangles in coverings of closed 2-orbifolds via ribbon categories

• Authors: Adam Klukowski
• arXiv: 2609.23652
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR); Probability (math.PR); Representation Theory (math.RT)

We describe what a typical covering of a closed 2-orbifold looks like on a small scale. Specifically, we prove that a uniformly random covering of degree n contains in expectation short closed geodesics, potentially pairs of nearby order-2 cone points, in general cone points of order m, and with high probability no small regions with more complicated topology. This generalises many results of Magee and Puder (2023) from orientable surfaces to possibly non-orientable orbifolds, and answers some of the questions raised by Puder and Zimhoni (2024).

More Versions of Real Link Floer Homology

• Authors: Yonghan Xiao
• arXiv: 2609.24354
• Comments: 64 pages in total= main part of 51 pages, 8 figures plus appendix jointly made with Zhenkun Li of 13 pages. Comments are welcome!
• Subjects: Geometric Topology (math.GT)

In this paper, we further develop the real link Floer homology defined by the first author. We introduce a new base-pointing convention that leads to a different version of real link Floer homology and show that this new theory is related to the old one by an exact triangle. We also define a real link Floer theory for multi-based strongly invertible links, which is a strong real Heegaard invariant, and take a first step toward a real link Floer TQFT. A computer implementation for the new theory via grid diagrams was written by Zhenkun Li. We also include an appendix containing real grid homology of more than 50 small knots.

Knots and the Sierpinski Tetrahedron

• Authors: Malors Espinosa
• arXiv: 2609.24545
• Subjects: Geometric Topology (math.GT); General Topology (math.GN)

In this paper we prove that there are infinitely many knots that cannot be embedded in the 1-skeletons of the finite iterations of the Sierpinski tetrahedron fractal. We do this by proving that such an embedding induces a sphere decomposition of weight at most 6. There are infinitely many knots with spherewidth greater than this.

Mapping class groups of negatively curved four-manifolds

• Authors: Weizhe Niu
• arXiv: 2609.24562
• Comments: 6 pages. Comments welcome
• Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG)

For every closed oriented smooth four-manifold $M$ admitting a metric of strictly negative sectional curvature, we prove that where $T(M)$ and denote the Torelli groups. This extends a theorem of Budney--Gabai for real hyperbolic manifolds.

Higher Barbell Diffeomorphisms and Watanabe's Clasper Surgery

• Authors: Xiayu Tan
• arXiv: 2609.24697
• Comments: 34 pages, 26 figures. Comments are very welcome!
• Subjects: Geometric Topology (math.GT)

Watanabe constructed many nontrivial family diffeomorphisms in , which were obtained by doing clasper surgeries on trivalent graphs. In this paper, we generalize the barbell diffeomorphism discovered by Budney and Gabai and construct a series of nontrivial elements in with the -cuff barbell, which we call higher barbell diffeomorphisms. We show that Watanabe's constructions can be realized as implanted higher barbell diffeomorphisms in if the trivalent graph is homologically nonzero in the graph complex. Combining this realization with Watanabe's detection theorem, we establish the nontriviality of these implanted higher barbell diffeomorphisms.

Nonsmoothable Embedded Surfaces in 4-Manifolds That Become Smoothable after a Single Stabilization

• Authors: Zuyi Zhang
• arXiv: 2609.24834
• Comments: 7 pages
• Subjects: Geometric Topology (math.GT)

In this paper, we find infinitely many examples of nonsmoothable embedded surfaces in a 4-manifold that become smoothable after taking the connected sum with S^2 x S^2. To the author's knowledge, no such examples have previously been discovered. In [CK25], Cha and Kim asked for the minimal number of stabilizations required to turn a nonsmoothable surface into a smoothable one. We provide sufficient conditions ensuring that this number is 1. In addition, we show that there is a nonsmoothable surface in a K3 surface such that, after a single stabilization, its topological isotopy class contains infinitely many smooth representatives that are pairwise not smoothly isotopic.

Trefoil Probabilities and Polyhedra

• Authors: Joseph Geisz; Chris Peterson; Clayton Shonkwiler
• arXiv: 2609.24860
• Comments: 12 pages, 6 figures
• Subjects: Geometric Topology (math.GT)

We determine the exact probability that the closed hexagonal polygon obtained by cyclically joining six independent points uniformly distributed on the unit sphere is knotted. The only possible nontrivial knot is a trefoil. Almost surely, the convex hull of the six points has one of two simplicial combinatorial types: a combinatorially regular octahedron and a combinatorially non-regular octahedron. We show that the straight-line complete graph on the six vertices of the hull of the combinatorially regular octahedral type admits exactly one unoriented trefoil Hamiltonian cycle. On the other hand, no cycle connecting vertices of the irregular octahedron can produce a trefoil. We then use stereographic projection to transform the probability of the regular hull type to a Sylvester-type problem for the planar beta-prime probability measure . Applying Stokes' theorem and the Blaschke-Petkantschin formula, we compute the expected squared $\mu$-content of a random triangle. This yields a regular octahedral probability of and a trefoil probability of .

Cross submissions

Expanders prevent Property (H)

• Authors: Bruno de Mendonç Braga; Chris Gartland; Gilles Lancien; Pavlos Motakis; Eva Pernecká; Thomas Schlumprecht
• arXiv: 2609.22416
• Cross-list: math.FA
• Comments: 7 pages. Lean verification provided
• Subjects: Functional Analysis (math.FA); Geometric Topology (math.GT); Metric Geometry (math.MG); Operator Algebras (math.OA)

We show that if a sequence of expander graphs equi-coarsely embeds into a Banach space, then this Banach space fails Kasparov and Yu's Property (H). Consequently, no Banach space with Property (H) can be coarsely universal for all countable groups. We provide a Lean verification of our results.

Chromatic symmetric functions for annular webs

• Authors: Jaeseong Oh; Seokbeom Yoon
• arXiv: 2609.22708
• Cross-list: math.CO
• Comments: 26 pages
• Subjects: Combinatorics (math.CO); Geometric Topology (math.GT)

We introduce a combinatorial definition of chromatic symmetric functions for annular webs. We prove their symmetry by constructing a web analogue of the Shareshian--Wachs involution and show that they coincide with the symmetric functions associated to annular webs via Turaev's isomorphism. We then derive explicit formulas for their hook Schur coefficients. We also introduce web LLT functions, whose hook Schur coefficients admit positive Laurent-polynomial formulas. These formulas yield a combinatorial expression for the coefficients of the HOMFLY--PT polynomial of an annular web.

A polynomial formula for the Albanese map of the Cartwright-Steger surface

• Authors: Zoltán Szabó
• arXiv: 2609.23124
• Cross-list: math.AG
• Comments: 14 pages
• Subjects: Algebraic Geometry (math.AG); Geometric Topology (math.GT)

The Albanese fibration of the Cartwright--Steger surface is a genus-$19$ Lefschetz fibration over a torus. We give an explicit formula for it: in the projective model of Borisov and Yeung, the map is represented by three homogeneous polynomials of degree $13$ with integer coefficients, and its target is the elliptic curve . A recognition theorem shows that it suffices to verify one polynomial identity on the surface. A coefficient bound allows us to establish the identity over $Q$ from computations modulo primes. We also determine exact coordinates for the three nodes and their images on the elliptic curve.

Sign components of diagonal superspace coinvariants

• Authors: Nicolle González; John Lentfer; Hanna Mularczyk
• arXiv: 2609.23297
• Cross-list: math.RT
• Comments: 29 pages, 1 figure. Comments welcome!
• Subjects: Representation Theory (math.RT); Combinatorics (math.CO); Geometric Topology (math.GT)

We prove the sign-isotypic components of the coinvariant rings and are isomorphic and show that the triply-graded multiplicity of this sign character is the Schröder polynomial , divided by . This settles the sign-character component of a conjecture of Zabrocki (2019) on a module for the Delta theorem and proves a conjecture of F. Bergeron (2020) on the multiplicity of the sign character of . Finally, using a result of Hogancamp (2017), we enhance a recent result of Gorsky--Mellit (2026) which relates the Khovanov--Rozansky homology of the -torus knot to , by showing that the associated Poincaré series for this knot can be computed from the sign component of .

Random groups in the square model have property (T) at densities over

• Authors: Damian Orlef; Tomasz Odrzygóźdź
• arXiv: 2609.24037
• Cross-list: math.GR
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT); Probability (math.PR)

A random group in the square model $\mathcal{Q}(n,d)$ is given by a presentation with $n$ generators and a uniformly random set of cyclically reduced relators of length 4 over them. We prove that if , then a random group in $\mathcal{Q}(n,d)$ has Kazhdan's property (T) asymptotically almost surely (a.a.s.). This provides a new construction of infinite word-hyperbolic Kazhdan groups when . Moreover, also for , we show that a random group in the square model a.a.s. has property for an increasing range of values of $p$, and property for any fixed uniformly curved Banach space $X$. These results lead to bounds on the conformal dimension of the boundary of a random group in the square model.

In the process we devise a method of applying the spectral link criteria for property (T), and other fixed point properties, to the groups presented by relations of length 4. In order to use it for the square model, we verify the required spectral gap hypothesis by the trace method, in the spirit of the work of Broder-Shamir.

Complete classification of algebraically primitive Teichm"uller curves in

• Authors: Myeongjae Lee
• arXiv: 2609.24587
• Cross-list: math.DS
• Comments: 16 pages, comments are welcome
• Subjects: Dynamical Systems (math.DS); Algebraic Geometry (math.AG); Geometric Topology (math.GT)

We give a complete classification of algebraically primitive Teichmüller curves in for . These curves are precisely those generated by the Veech $2p$-gons, where is prime.

Non-maximal quasi-isometric -representations via alternating surfaces in complex pseudo-hyperbolic spaces

• Authors: Qiongling Li; Junming Zhang
• arXiv: 2609.24672
• Cross-list: math.DG
• Comments: 55 pages, including two appendices and a declaration of AI use, comments are very welcome
• Subjects: Differential Geometry (math.DG); Geometric Topology (math.GT)

For every admissible non-maximal Toledo invariant $t$, we construct a locus of irreducible representations of a closed genus-$g$ surface group into with Toledo invariant $t$ whose orbit maps are quasi-isometric embeddings. These loci have real codimension or in the character variety depending on the Toledo value. Our construction uses $4$-cyclic Higgs bundles and their correspondence with $\partial$-alternating surfaces in complex pseudo-hyperbolic spaces. The associated equivariant minimal maps into the symmetric space are bi-Lipschitz embeddings. We further construct cocompact domains of discontinuity in the Shilov boundary. Their quotients are smooth fiber bundles over the surface with fiber homeomorphic to .

Fixed point properties and cohomology of Banach representations of arithmetic groups

• Authors: Saar Bader; Shaked Bader; Uri Bader; Roman Sauer
• arXiv: 2609.24951
• Cross-list: math.GR
• Comments: 45 pages
• Subjects: Group Theory (math.GR); Algebraic Topology (math.AT); Geometric Topology (math.GT)

We study fixed point theorems for actions of lattices of semisimple groups. They are deduced from vanishing results for the group cohomology of -representations. We show that for lattices in simple groups of higher rank, the cohomology with -coefficients vanishes below the rank whenever there are no invariant vectors. As a corollary of the vanishing for -coefficients, we obtain that every action on an acyclic simplicial complex of dimension lower than the rank has a finite orbit. This in particular proves a conjecture by Farb. The -vanishing below the rank proves a conjecture by Gromov regarding -cohomology of symmetric spaces.