Brieskorn submanifolds, Local moves on knots, and knot products

TL;DR

This paper characterizes high-dimensional Brieskorn submanifolds and their relation to Seifert matrices, local moves on knots, and knot products, providing new equivalence criteria and extending known results to specific cases.

Contribution

It establishes a new classification criterion for Brieskorn submanifolds using Seifert matrices and explores their relation to local moves and knot products in higher dimensions.

Findings

Knot equivalence characterized by Seifert matrix $(-1)^p$-$S$-equivalence.

Pass-move equivalence of 1-links related to knot products with Hopf links.

Two-fold cyclic suspension commutes with twist moves for certain spherical knots.

Abstract

We prove the following: Let $2p + 1$ be no less than 5 and $p$ be a natural number. Let $K$ and $J$ be closed, oriented, $(2p+1)$-dimensional connected, $(p-1)$-connected, simple submanifolds of the standard $(2p+3)$-sphere. Then $K$ is equivalent to $J$ 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 $2p+1=3$ case. This result implies one of our main results: Let $\mu$ be a natural number. A 1-link $A$ is pass-move equivalent to a 1-link $B$ if and only if the knot product of $A$ and $\mu$ copies of the Hopf link is $(2\mu+1, 2\mu+1)$-pass-move equivalent to that of B and $\mu$ copies of the Hopf link. It also implies the other of them: Two-fold cyclic suspension commutes with the performance of the twist move for spherical $(2k+1)$-knots ($2k+1>4$). Furthemroe we prove the following: Let $2p+1$ be no less than 5 and p be a natural number. Let $K$ be a closed oriented $(2p+1)$-dimensionalsubmanifold of the standard $(2p+3)$-sphere. 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 Kauffman-Neumann-type, or a KN-type (See the body of the paper for a definition.) We also discuss the $2p+1=3$ case.