The Conway polynomial and amphicheiral knots

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 [Formula: see text] for some [Formula: see text]. 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 [Formula: see text] in the ring [Formula: see text]. 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.