Brieskorn submanifolds, local moves on knots, and knot products
Abstract
We first prove the following: Let p≥2p≥2 and p∈ℕ. 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∈ℕ. 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 J⊗K⋯⊗K︸μ. 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∈ℕ. 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.
