The Conway Polynomial and Amphicheiral Knots

TL;DR

This paper investigates the factorization properties of the Conway Polynomial for amphicheiral knots, proving a conjecture for a class of periodically amphicheiral knots and providing counterexamples to previous conjectures.

Contribution

It proves the Conway polynomial splitting conjecture in the ring [z] for periodically amphicheiral knots constructed from braids, and presents counterexamples to earlier conjectures.

Findings

Conjecture holds for periodically amphicheiral braid knots.

Counterexamples to leading coefficient conjectures.

Provides new insights into Conway Polynomial factorization.

Abstract

According to work of Hartley and Kawauchi in 1979 and 1980, the Conway Polynomial of all negative amphicheiral knots and strongly positive amphicheiral knots factors as $\phi(z)\phi(-z)$ for some $\phi(z)\in\mathbb Z[z]$. Moreover, a 2012 example due to Ermotti, Hongler and Weber shows that this is not true for general amphicheiral knots. On the other hand, in 2006 the first author made a conjecture equivalent to saying that the Conway polynomial of all amphicheiral knots splits as $\phi(z)\phi(-z)$ in the ring $\mathbb Z_4[z]$. In this paper, we establish this conjecture for all periodically amphicheiral knots built from braids, where the period preserves the braid structure. We also give counterexamples to conjectures on the leading coefficient of the Conway polynomial of an amphicheiral knot due to Stoimenow.