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

New submissions

A proof of the mod 4 Kawauchi Conjecture

• Authors: Jim Conant
• arXiv: 2607.18655
• Subjects: Geometric Topology (math.GT)

Kawauchi conjectured that the Conway polynomial of an amphicheiral knot factors as for some integer polynomial $z$. In joint work with Hartley, he showed this was true for strongly amphicheiral knots, and Hartley used the JSJ decomposition of the knot exterior to generalize to all negative amphicheiral knots. Ermotti--Hongler--Weber were the first to publish a counterexample to the general case. Independently, in 2006 the author had conjectured a statement which is equivalent to the statement that for amphicheiral knots, based on certain patterns he noticed in finite type invariants. In this paper we prove this mod 4 version of Kawauchi's original conjecture. This proof was produced with the help of Claude Fable 5.

The complex projective plane as a ball quotient

• Authors: Cindy Tan
• arXiv: 2607.18710
• Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG); Differential Geometry (math.DG)

In 1986, Deligne and Mostow constructed a ball quotient biholomorphic to the complex projective plane whose branch locus is a line arrangement. In this paper, we show that if is realized as a ball quotient whose branch divisor $D$ is an arrangement of smooth pairwise normal-crossing curves, then the orbifold is isomorphic to either the Deligne-Mostow example or a certain degree 9 cover of it. This classification of "ball quotient structures" on generalizes the case due to Poincaré.

Cross submissions

Relative free splitting and free factor complexes: An overview

• Authors: Michael Handel; Lee Mosher
• arXiv: 2607.19249
• Cross-list from: Group Theory (math.GR)
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT)
• Comments: 20 pages + references

For any group $\Gamma$ and any free factor system~$\mathcal{A}$ of $\Gamma$, the relative outer automorphism group acts naturally on the relative free splitting complex and on the complex of relative free factor systems , generalizing the well known actions of on the absolute free splitting complex and the absolute free factor complex of the rank~$n$ free group .

This overview summarizes a three part work regarding the large scale geometry of and and the geometric dynamics of the actions on these complexes by elements of . In Part I arXiv:1407.3508 we prove hyperbolicity of and of . In Parts II and III arXiv:2212.09907, arXiv:2503.07532 we study the relation between the geometric dynamics of an element of and the dynamics of its relative train track representatives. The main tool in Part II is the \emph{Two Over All Theorem}, expressing an exponential flaring property of Stallings fold paths in . The main tools in Part III are \emph{filling paths}, used to formulate and prove a strong version of the \emph{Two Over All Theorem}.

Degenerations of the complex projective plane with only rational singularities

• Authors: Marcos Canedo; Giancarlo Urzúa
• arXiv: 2607.19348
• Cross-list from: Algebraic Geometry (math.AG)
• Subjects: Algebraic Geometry (math.AG); Geometric Topology (math.GT); Symplectic Geometry (math.SG)

Wahl's conjecture states that two-dimensional singularities admitting a rational homology disk smoothing are weighted homogeneous. Assuming the conjecture, we classify all normal degenerations of the complex projective plane with only rational singularities. They are precisely the surfaces classified by Manetti and Hacking--Prokhorov, which are controlled by the Markov equation, together with six new degenerations containing four non-log canonical singularities.