math.GT daily digest: 4 new submissions for 22 October 2026

New submissions

Polynomial Invariants of Pseudo-Anosov Maps (Addendum)

• Authors: Merrick Dodge, Keiko Kawamuro
• arXiv: 2610.28959
• Comments: 30 pages, 17 figures
• Subjects: Geometric Topology (math.GT)

We show that homology, puncture, and symplectic polynomials of a pseudo-Anosov map arXiv:1001.5094 are invariants of the mapping class, as well as of the conjugacy class.

Asymptotics of Masur-Veech volume of strata of quadratic differentials with many simple poles

• Authors: Kasra Rafi, Anton Zorich
• arXiv: 2610.28963
• Subjects: Geometric Topology (math.GT)

We provide large $n$ asymptotics of the Masur-Veech volumes for strata of quadratic differentials of the form , where all positive integers are fixed and . Our formulae are particularly explicit in genus $1$. For the two series of non-varying strata and we give simple closed formulae for the volumes for any $n$.

The geometric combinatorics of $k$-shuffle braid groups

• Authors: Chloé Postel-Vinay
• arXiv: 2610.28986
• Comments: This is the first paper in a series of two, with the second one to be uploaded later in the week
• Subjects: Geometric Topology (math.GT); Combinatorics (math.CO); Group Theory (math.GR)

This paper is the first in a series of two which aims at introducing a new family of groups called $k$-shuffle braid groups, denoted . These groups have two intrinsically related stories. They were first observed by Calegari in arXiv:2105.11265 by looking at spaces of complex monic polynomials of degree $n$ whose critical values lie in . The $k$-shuffle braid groups are the fundamental groups of those spaces, which correspond geometrically to complements of $k$ translates of the discriminant variety. The second story starts from the poset of $k$-shuffle partitions, which were introduced by Drew Armstrong and explored in depth in his thesis arXiv:math/0611106. In this paper, we focus on the combinatorial aspects of $k$-shuffle braid groups starting from the poset of $k$-shuffle non-crossing partitions. We build a CW-complex on which the groups act freely, properly discontinuously and cocompactly, which we show is CAT(0) for dimension up to 7. For higher dimension, we show that the CAT(0) property is implied by the conjecture of Brady and McCammond that the dual braid complex is CAT(0) (arXiv:0909.4778).

Affine $G$-multibiquandle modules for colored links I: representability of coloring functors and color-profile polynomials

• Authors: Sang Youl Lee
• arXiv: 2610.29011
• Comments: 54 pages, 3 figures
• Subjects: Geometric Topology (math.GT); Combinatorics (math.CO)

Let $R$ be a commutative ring with identity, let $G$ be an abelian group, and fix . For each component-labeled oriented link diagram $(D,\sigma )$, we construct a finitely presented $R[G]$-module with a universal affine coloring . Every affine coloring with values in an $R[G]$-module factors uniquely through by a module homomorphism. Thus the module represents the affine coloring functor. Each specified colored Reidemeister move induces a unique module isomorphism compatible with coloring transport, and the module isomorphism class is a colored-link invariant. The representing modules and universal colorings commute with arbitrary scalar extension, without a flatness assumption. For classical links, simultaneous rescaling of the component labels removes the common deformation unit.

For any multibiquandle, associated-biquandle colorings decompose into fibers indexed by component labelings. For finite targets and palettes, we define a color-profile polynomial recording the joint distribution of fiber sizes and numbers of palette labels used, and prove its invariance under oriented link isotopy. In the affine case, the representing modules determine these fibers, and their specialized dimensions determine the coloring counts over finite fields. The color-profile polynomial is not determined, in general, by the full unweighted distribution of coloring-fiber sizes. It therefore retains information beyond the associated-biquandle coloring count.