math.GT daily digest: 4 submissions for 18 September 2026

New submissions

Combinatorial Goussarov-Polyak-Viro Formulas for the Linking Number and Low Degree Coefficients of the Conway Polynomial

• Authors: Nancy Scherich; Nathaniel Song
• arXiv: 2609.20413
• Subjects: Geometric Topology (math.GT)

The linking number and coefficients of the Conway polynomial are two examples of finite type invariants. Goussarov-Polyak-Viro proved that any finite type knot invariant can be computed in a two step process, where the second step is referred to as the GPV map for the invariant. This paper computes the GPV maps for the linking number and low-degree coefficients of the Conway Polynomial. Chmutov-Khoury-Rossi gave one description of the GPV maps for coefficients of the Conway polynomial in terms of arrow diagrams and state-sum calculations. We take a much more grounded approach using fundamental linear algebra to describe the GPV maps in terms of a basis for the vector space of Gauss diagrams.

Cross submissions

Chow Vanishing and Motives of Cluster Varieties

• Authors: Josephine Hlavinka
• arXiv: 2609.19744
• Cross-list: math.AG
• Comments: 23 pages, 2 figures. Comments welcome!
• Subjects: Algebraic Geometry (math.AG); Combinatorics (math.CO); Geometric Topology (math.GT); Representation Theory (math.RT)

We prove that the integral Chow groups and mixed Hodge degree cohomology groups of really full rank (RFR) sink-recurrent cluster varieties vanish for . In particular this applies to braid varieties and open Richardson varieties in any Lie type. Our main tool is the construction of a stratification of any RFR sink-recurrent cluster variety $\mathcal{A}(\Sigma )$ into (affine spaces times) RFR sink-recurrent cluster varieties of seeds with fewer mutable vertices than $\Sigma$.

We employ the theory of Voevodsky motives, and towards this end we prove that the cycle class maps are isomorphisms onto the lowest-weight part of rational Borel-Moore homology for any mixed Tate variety over a number field. We then show that RFR sink-recurrent cluster varieties have mixed Tate and, in fact, split motives. Finally, we use our results to deduce vanishing theorems about the Khovanov-Rozansky homology groups of closures of positive braids and generation properties of the cohomology of closed Richardson, projected Richardson, and brick varieties.

Representation stability of string links and manifold links

• Authors: Filipp Buryak
• arXiv: 2609.20470
• Cross-list: math.AT
• Subjects: Algebraic Topology (math.AT); Geometric Topology (math.GT)

We prove representation stability for rational cohomology of spaces of string links and manifold links in Euclidean space as the number of links grows. For string links we consider the component of the standard embedding of disjoint copies of into , where and . For manifold links we consider the component determined by disjoint copies of a fixed embedding of a closed smooth $m$-manifold into , where . In both cases, the rational cohomology groups form finitely generated -modules in each degree with generation arity at most in degree $n$. We also obtain explicit generation bounds for the rational homotopy groups. The proofs combine geometric constructions of structure maps, hairy graph complex models for embeddings modulo immersions and a functorial framework that allows generation bounds to be established after forgetting the symmetric group actions.

The Hurwitz existence problem in prime degree

• Authors: Jijian Song; Hailin Wen; Zebao Zhang
• arXiv: 2609.20572
• Cross-list: math.AG
• Subjects: Algebraic Geometry (math.AG); Geometric Topology (math.GT)

Let $p$ be a prime. We prove that every compatible branch datum of degree $p$ over the sphere is realizable by a connected branched cover. The three-point case is constructed in residue characteristic $p$. Henrio's moment theorem supplies the distinct-point moment solutions from which we construct a special primitive tail for each prescribed partition; a second application underlies the new tail required by a positive source genus. These tails are joined by a logarithmic deformation datum and embedded in one subgroup of $S\_p$ containing a common regular subgroup of order $p$. Wewers's lifting theorem produces a three-point Galois cover in characteristic zero. The quotient by a point stabilizer has degree $p$ and the prescribed three ramification profiles. The fusion and realization results of Edmonds--Kulkarni--Stong then give the assertion for an arbitrary number of branch values. As consequences, the connected prime-degree Hurwitz potential has full support on the Riemann--Hurwitz locus, every corresponding connected relative Gromov--Witten invariant of is nonzero, the two-relative-point disconnected sector with even completed-cycle orders at most $p$ is strictly positive subject to the dimension constraint, and the connected transposition sector is strictly positive.