On the Kawauchi conjecture about the Conway polynomial of achiral knots

TL;DR

This paper provides a counterexample to the Kawauchi conjecture, showing that the Conway polynomial of certain achiral knots does not always satisfy the splitting property, and explores its relation to knot decompositions.

Contribution

It presents a counterexample to the Kawauchi conjecture and links the Conway polynomial's properties to the Bonahon-Siebenmann decomposition of achiral, alternating knots.

Findings

Counterexample disproves the conjecture for some alternating knots.

Kawauchi conjecture holds for quasi-arborescent knots.

Counterexamples are quasi-polyhedral knots.

Abstract

We give a counterexample to the Kawauchi conjecture on the Conway polynomial of achiral knots which asserts that the Conway polynomial $C(z)$ of an achiral knot satisfies the splitting property $C(z)=F(z)F(-z)$ for a polynomial $F(z)$ with integer coefficients. We show that the Bonahon-Siebenmann decomposition of an achiral and alternating knot is reflected in the Conway polynomial. More explicitly, the Kawauchi conjecture is true for quasi-arborescent knots and counterexamples in the class of alternating knots must be quasi-polyhedral.