Brieskorn submanifolds, local moves on knots, and knot products

Abstract

We first prove the following: Let [Formula: see text] and [Formula: see text]. Let [Formula: see text] and [Formula: see text] be closed, oriented, [Formula: see text]-dimensional [Formula: see text]-connected, simple submanifolds of [Formula: see text]. Then [Formula: see text] and [Formula: see text] are isotopic if and only if a Seifert matrix associated with a simple Seifert hypersurface for [Formula: see text] is [Formula: see text]-[Formula: see text]-equivalent to that for [Formula: see text]. We also discuss the [Formula: see text] case. This result implies one of our main results: Let [Formula: see text]. A 1-link [Formula: see text] is pass-equivalent to a 1-link [Formula: see text] if and only if [Formula: see text] is [Formula: see text]-pass-equivalent to [Formula: see text]. Here, [Formula: see text] means the knot product of [Formula: see text] and [Formula: see text], and [Formula: see text] means [Formula: see text]. 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 [Formula: see text]-knots. See the body for the precise statement. Furthermore, it implies the following: Let [Formula: see text] and [Formula: see text]. Let [Formula: see text] be a closed oriented [Formula: see text]-submanifold of [Formula: see text]. Then [Formula: see text] is a Brieskorn submanifold if and only if [Formula: see text] is [Formula: see text]-connected, simple and has a [Formula: see text]-Seifert matrix associated with a simple Seifert hypersurface that is [Formula: see text]-[Formula: see text]-equivalent to a [Formula: see text]-type (see the body of the paper for a definition). We also discuss the [Formula: see text] case.