math.GT daily digest: 11 submissions for 12 October 2026

New submissions

Symmetries of colored Khovanov-Rozansky homology

• Authors: Alexis Guérin, Felix Roz
• arXiv: 2610.13210
• Comments: 38 pages, many figures
• Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA)

We construct an action of the positive half of the Witt algebra on colored -Khovanov-Rozansky homology and prove that this action is compatible with the functoriality of colored link homology. To establish this result, we use Bar-Natan's framework of canopolies to formulate and prove the relevant statements at the level of tangles.

Recognizing oriented knots from finite quotients with one marked element

• Authors: Indranil Biswas, Shrinit Singh
• arXiv: 2610.13260
• Comments: 33 pages
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

For an oriented tame knot $K$, let be its group, with positive meridian and preferred longitude . We prove that the profinite completion , marked by the conjugacy class of , determines $K$ up to oriented equivalence. In fact, every isomorphism preserving this mark preserves the meridian and longitude with the same conjugator. The proof combines Xu's genuineness theorem with profinite JSJ theory and simultaneous boundary realization, and applies to satellite and composite knots. Consequently, Eisermann's colouring polynomials over all finite groups determine oriented knots.

Thurston's Legacy at the Interface of Surface Dynamics, Train Tracks, and Arithmetic

• Authors: Ahmad Rafiqi, Ayberk Zeytin
• arXiv: 2610.13347
• Comments: 36 pages, 1 figure
• Subjects: Geometric Topology (math.GT); Number Theory (math.NT)

In this chapter, we discuss the interaction between low-dimensional dynamics, symbolic dynamics, and geometric topology, focusing on the algebraic characterization and asymptotic behaviour of stretch factors or equivalently pseudo-Anosov dilatations. We first review the algebraic foundations linking Perron--Frobenius theory, multi-variable Euclidean algorithms, and Lind's characterization of topological entropy for shifts of finite type, emphasizing recent constraints on the Perron--Frobenius degree of expansion constants. We then relate these dynamical invariants to the combinatorial world, examining train tracks for surfaces and automorphisms of free groups. Finally, we briefly review the relevance of fibered 3-manifolds, detailing how the Thurston norm, Fried's concavity theorem over fibered cones, and McMullen's Teichmüller polynomial come into play.

Finding the Birman-Series set

• Authors: Peter Buser, Hugo Parlier
• arXiv: 2610.13645
• Comments: 18 pages, 28 figures
• Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG)

By a celebrated result of Birman and Series, the set of simple complete geodesics form a nowhere dense subset of a hyperbolic surface. We show that a segment of finite length is in a (small) neighborhood of this set if it can be "extended simply" in both directions for long enough. We deduce that, for large enough genus $g$, there exist surfaces with a Birman-Series set that is roughly \frac{1}{g^{\sfrac{2}{3}}}!-dense.

Linearity of Graph Manifold Groups

• Authors: Daniel T. Wise
• arXiv: 2610.13801
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

We complete the proof that every compact 3-manifold group is linear. Specifically, we construct a faithful representation in GL5(Q[u]) for a finite-index subgroup of every closed non-geometric graph-manifold group.

Cross submissions

Self-intersecting sections of polar actions on simply connected manifolds

• Authors: Stephan Wiesendorf
• arXiv: 2610.12446
• Cross-list: math.DG
• Comments: 5 pages
• Subjects: Differential Geometry (math.DG); Geometric Topology (math.GT)

We construct a polar action of $\mathrm{SU}(3)$ on a closed simply connected $11$-manifold with a compact three-dimensional section that is not injectively immersed. The section is injective over the regular part and has two distinct local branches along a singular stratum. All isotropy groups are connected, and all orbits are simply connected. This gives a negative answer to the injectivity question raised by Grove and Ziller.

Singular Levi-flats and holomorphic foliations on projective spaces

• Authors: Bruno Scárdua
• arXiv: 2610.12497
• Cross-list: math.CV
• Subjects: Complex Variables (math.CV); Geometric Topology (math.GT)

We study codimension-one holomorphic foliations admitting invariant singular real-analytic Levi-flat varieties. Our main problem is to determine when the local meromorphic first integrals furnished by the Levi-flat geometry can be continued coherently and globalized. We introduce a pencil-singularity condition and a projective compatibility condition for local meromorphic parameters. In the projective plane these yield a characterization of foliations admitting a primitive rational first integral, while in higher-dimensional projective space they give a rational integrability criterion under a generic local-product hypothesis along the dicritical locus. We also obtain semi-global and Stein versions of these results. In a different direction, we study foliations admitting infinitely many invariant Levi-flat varieties and prove a Darboux-type alternative: either the foliation admits a rational first integral, or infinitely many Levi-flats contain a common invariant algebraic curve whose virtual holonomy is abelian. Under additional assumptions on the singularities, this leads to the existence of a closed rational, and in the absence of saddle-nodes a closed logarithmic, defining $1$-form.

Chern classes of Levi non-degenerate CR manifolds

• Authors: Yuya Takeuchi
• arXiv: 2610.12785
• Cross-list: math.CV
• Comments: 13 pages
• Subjects: Complex Variables (math.CV); Differential Geometry (math.DG); Geometric Topology (math.GT)

We construct closed Levi non-degenerate CR manifolds with non-vanishing Chern monomials. For every and , there exists a -dimensional example of signature on which all Chern monomials with positive indices at most $m$ and total weight at most are non-zero in cohomology with real coefficients. Taking gives examples with . For $n=4$ or , the latter examples show that the Chern-class vanishing theorem for strictly pseudoconvex CR manifolds does not extend to the Levi non-degenerate case. In these dimensions, their underlying manifolds admit no strictly pseudoconvex CR structures. The construction uses circle bundles over products of integral pseudo-Kähler manifolds.

Quasipositive Braids, Presentations, and Garside Structures

• Authors: Jacob Guynee
• arXiv: 2610.12808
• Cross-list: math.GR
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT)

We provide a presentation of the braid group in which the generating set is the set of all positive half-twists, the relations are balanced positive relations, and every relation is conjugate to one in a finite list. We then prove that for any such presentation, its positive monoid is not isomorphic to the quasipositive braid monoid. Finally, we show that the quasipositive braid monoid has no proper Garside family. These results settle a problem of Calegari from the K3 problems list.

Critical Point Theory for the Distance Function on PL-Spaces

• Authors: Dirk Siersma
• arXiv: 2610.13297
• Cross-list: math.AT
• Comments: 15 pages, 1 figure
• Subjects: Algebraic Topology (math.AT); Geometric Topology (math.GT)

We study the (typical) singularities of the distance function from a point in the ambient space to a polytopal complex in Euclidean space. By means of the Goresky-MacPherson splitting we generalize the existing linear theory to the quadratic distance function. Critical points occur on `selected' strata of the PL-space, where normal Morse Data determine the change of topology. On the homology level there exist indices with multiplicities, combined in a Morse vector. We focus mainly on PL-manifolds. We give bounds, depending on the dimension of the strata. In low dimensions we describe singularity types. At the end we discuss relations with Discrete Morse Theory.

On some rigidity results for high-dimensional confoliations

• Authors: Fabio Gironella, Seungook Yu
• arXiv: 2610.13927
• Cross-list: math.SG
• Comments: V1: 49 pages. Comments are welcome!
• Subjects: Symplectic Geometry (math.SG); Geometric Topology (math.GT)

The main result is an obstruction to the presence of contact regions for c-symplectic confoliations in any odd ambient dimension. As a first application, we prove a 5-dimensional analogue (under stronger assumptions) of a Reeb stability theorem for confoliations in dimension 3 due to Eliashberg and Thurston. Refining work of the first author with Cardona, we further give two new notions of contact approximations of c-symplectic confoliations in any odd ambient dimension generalizing the 3-dimensional notion of Eliashberg-Thurston. As a second application of our main theorem, we then give an example, in any ambient odd dimension, of a c-symplectic (and in fact strong symplectic) foliation that cannot be contact approximated, in either of these two senses. This relies on the property, of independent interest, that contact structures approximating (in these senses) a weakly c-symplectic confoliation are also weakly fillable. The proofs of all these statements rely on pseudo-holomorphic curve theory. Lastly, we describe several natural classes of examples of these types of contact approximations of c-symplectic confoliations.