Brieskorn submanifolds, local moves on knots, and knot products

Abstract

We first prove the following: Let p ≥ 2 and p ∈ N. Let K and J be closed, oriented, (2p + 1)-dimensional connected, (p − 1)-connected, simple submanifolds of S 2p+3 . Then K and J are isotopic if and only if a Seifert matrix associated with a simple Seifert hypersurface for K is (−1) p -S-equivalent to that for J. We also discuss the p = 1 case.This result implies one of our main results:It also implies the other of our main results: We strengthen the authors' old result that two-fold cyclic suspension commutes with the performance of the twist move for spherical (2k + 1)-knots. See the body for the precise statement.Furthermore it implies the following: Let p ≥ 2 and p ∈ N. Let K be a closed oriented (2p + 1)-submanifold of S 2p+3 . Then K is a Brieskorn submanifold if and only if K is connected, (p − 1)-connected, simple and has a (p + 1)-Seifert matrix associated with a simple Seifert hypersurface that is (−1) p -S-equivalent to a KN -type (see the body of the paper for a definition). We also discuss the p = 1 case.(2) Let K and J be closed oriented 3-dimensional simple submanifolds of S 5 . Then the following two statements are equivalent. (i) K is isotopic to J.(ii) There is a simple Seifert hypersurface V K (respectively, V J ) for K (respectively, J) with the following properties: There is an orientation preserving diffeomorphism map f : V K → V J . V K (respectively, V J