Local moves on knots and products of knots II

Abstract

We use the terms, knot product and local move, as defined in the text of the paper. Let $n$ be an integer$\geqq3$. Let $\mathcal S_n$ be the set of simple spherical $n$-knots in $S^{n+2}$. Let $m$ be an integer$\geqq4$. We prove that the map $j:\mathcal S_{2m}\to\mathcal S_{2m+4}$ is bijective, where $j(K)=K\otimes$Hopf, and Hopf denotes the Hopf link. Let $J$ and $K$ be 1-links in $S^3$. Suppose that $J$ is obtained from $K$ by a single pass-move, which is a local-move on 1-links. Let $k$ be a positive integer. Let $P\otimes^kQ$ denote the knot product $P\otimes\underbrace{Q\otimes...\otimes Q}_k$. We prove the following: The $(4k+1)$-dimensional submanifold $J\otimes^k{\rm Hopf}$ $\subset S^{4k+3}$ is obtained from $K\otimes^k{\rm Hopf}$ by a single $(2k+1,2k+1)$-pass-move, which is a local-move on $(4k+1)$-submanifolds contained in $S^{4k+3}$. See the body of the paper for the definitions of all local moves in this abstract. We prove the following: Let $a,b,a',b'$ and $k$ be positive integers. If the $(a,b)$ torus link is pass-move equivalent to the $(a',b')$ torus link, then the Brieskorn manifold $\Sigma(a,b,\underbrace{2,...,2}_{2k})$ is diffeomorphic to $\Sigma(a',b',\underbrace{2,...,2}_{2k})$ as abstract manifolds. Let $J$ and $K$ be (not necessarily connected or spherical) 2-dimensional closed oriented submanifolds in $S^4$. Suppose that $J$ is obtained from $K$ by a single ribbon-move, which is a local-move on 2-dimensional submanifolds contained in $S^4$. Let $k$ be an integer$\geq2$. We prove the following: The $(4k+2)$-submanifold $J\otimes^k{\rm Hopf}$ $\subset S^{4k+4}$ is obtained from $K\otimes^k{\rm Hopf}$ by a single $(2k+1,2k+2)$-pass-move, which is a local-move on $(4k+2)$-dimensional submanifolds contained in $S^{4k+4}$.