Brieskorn submanifolds, local moves on knots, and knot products

We first prove the following: Let p ≥ 2 and p ≥. Let K and J be closed, oriented, (2p + 1)-dimensional (p-1)-connected, simple submanifolds of S2p+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: Let μ ≥. A 1-link A is pass-equivalent to a 1-link B if and only if A μHopf is (2μ + 1, 2μ + 1)-pass-equivalent to B μHopf. Here, J K means the knot product of J and K, and J μK means JK⋯ Kïμ. See the body of the paper for the definition of knot products. It also implies the other 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 ≥. Let K be a closed oriented (2p + 1)-submanifold of S2p+3. Then K is a Brieskorn submanifold if and only if K is (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.