A proof of the mod 4 Kawauchi Conjecture

Jim Conant

Abstract

Kawauchi conjectured that the Conway polynomial of an amphicheiral knot factors as $\nabla_K(z)=f(z)f(-z)$ for some integer polynomial $f(z)$. 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 $\nabla_K \equiv f(z)f(-z) \pmod 4$ 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 as a consequence of a stronger integral statement. The mathematical content of this paper was produced with the help of Claude Fable 5.

Disclosure

“terns he noticed in finite type invariants. In this paper we prove this mod 4 version of Kawauchi's original conjecture as a consequence of a stronger integral statement. The mathematical content of this paper was produced with the help of Claude Fable 5.”
arXiv metadata: abstract

Classification: Substantial mathematical content or result generation
Multiplier: 10
Verified: 2026-08-23T19:00:19.001551Z

Structural counts

Pages 23 pdf

Theorems 12 source

Lemmas 13 source

Propositions 4 source

Corollaries 4 source

Definitions 3 source

Displayed equations 78 source

Bibliography entries 17 source

Appendix pages 5 estimated

Count notes

Source counts use the expanded primary TeX file main.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.