On the Reversibility of Twist-spun Knots

Litherland has shown that if a knot is (+)-amphicheiral then its m-twist-spin is reversible. We show that, for classical knots, in many cases the converse holds. The irreversibility (sometimes called noninvertibility) of certain twist-spun knots has been established by Ruberman [10], using the Farber-Levine linking pairing and the Casson-Gordon invariants. More recently, alternative proofs of the irreversibility of the 2-twist-spun trefoil have been given by Carter, Jelsovsky, Kamada, Langford and Saito [4], and by Rourke and Sanderson [9], using quandle cohomology and the homotopy theory of racks, respectively. Similar methods have been used by Satoh [11] to prove the irreversibility of certain other twist-spun torus knots. In the present note we use more geometric methods to prove the following more general result. Theorem 1. (1) The 2-twist-spin of a rational knot κ is reversible if and only if κ is amphicheiral. (2) If m, p, q are > 1 then the m-twist-spin of the (p, q) torus knot is irreversible. (3) If m ≥ 3 then the m-twist-spin of a hyperbolic knot κ is reversible if and only if κ is (+)-amphicheiral. Since a rational knot is either hyperbolic or a (2, q) torus knot, we obtain the following corollary. Corollary 2. If m ≥ 2 then the m-twist-spin of a rational knot κ is reversible if and only if κ is amphicheiral. The “if” directions in parts (1) and (3) of Theorem 1 are due to Litherland [5], who shows that the m-twist-spin of a (+)-amphicheiral knot is always reversible. We work in the PL category. A knot κ (more precisely, an n-knot) is a locally flat oriented pair (S,K), where K is homeomorphic to S. (The knots in Theorem 1 ∗Partially supported by TARP grant 003658-0519-2001.