Conant’s mod-$4$ Kawauchi conjecture

The conjecture asks whether every amphicheiral knot has a Conway polynomial congruent modulo $4$ to $f(z)f(-z)$ for some $f(z)\in\mathbb{Z}[z]$. Conant proposed this mod-$4$ version in $2006$, after the stronger integral factorization statement was shown to fail.

Known results

• Kawauchi and Hartley proved the exact integral factorization for strongly amphicheiral knots.
• Hartley extended it to all negative amphicheiral knots using the JSJ decomposition.
• Ermotti–Hongler–Weber produced a counterexample to unrestricted integral factorization; it still satisfies the mod-$4$ factorization.

2026 proof

Jim Conant’s arXiv paper proves that for every amphicheiral knot there is an $f(z)\in\mathbb{Z}[z]$ with $\nabla_K(z)\equiv f(z)f(-z)\pmod{4}$. A separate announcement says the result was obtained using Fable 5 under human direction and verification.

Current status (as of July 2026): The mod-$4$ conjecture is settled for all amphicheiral knots by the arXiv proof; the stronger integral factorization statement remains false.