Nonamphicheiral codimension 2 knots

Abstract

An n-knot (Sn+2,Sn) is said to be amphicheiral if there is an orientation-reversing autohomeomorphism of Sn+2 which leaves Sn invariant as a set. An n-knot is said to be invertible if there is an orientation-preserving autohomeomorphism of Sn+2 whose restriction to Sn is an orientation-reversing autohomeomorphism of Sn. The authors prove that for any integer n there are smooth n-knots which are neither amphicheiral nor invertible. Actually, they prove it for n≥2, referring to the paper of H. F. Trotter [Topology 2 (1963), 275–280; errata, MR 30, p. 1205] for the case n=1. The methods employed are mainly algebraic, involving for example the duality pairings of R. C. Blanchfield and J. Levine, and in most cases the work of previous authors is used to guarantee the existence of knots with the desired algebraic properties.