Narrow Results

Recently Kearton showed that any Seifert matrix of a knot is S-equivalent to the Seifert matrix of a prime knot. We show in this note that such a matrix is in fact S-equivalent to the Seifert matrix of a hyperbolic knot. This result follows from reinterpreting this problem in terms of Blanchfield pairings and by applying results of Kawauchi.

We first prove the following: Let $p\ge 2$ and $p\in \mathbb{N}$. Let $K$ and $J$ be closed, oriented, $(2p+1)$-dimensional $(p-1)$-connected, simple submanifolds of $S^{2p+3}$. 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 $\mu \in \mathbb{N}$. A 1-link $A$ is pass-equivalent to a 1-link $B$ if and only if $A{\otimes }^{\mu }\mathrm{\text{Hopf}}$ is $(2\mu +1,2\mu +1)$-pass-equivalent to $B{\otimes }^{\mu }\mathrm{\text{Hopf}}$. Here, $J\otimes K$ means the knot product of $J$ and $K$, and $J{\otimes }^{\mu }K$ means $J\underset{\mu }{\underbrace{\otimes K\cdots \otimes K}}$. 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 $p\ge 2$ and $p\in \mathbb{N}$. Let $K$ be a closed oriented $(2p+1)$-submanifold of $S^{2p+3}$. 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.

We use the terms, knot product and local-move, as defined in the text of this paper. Let $n$ be an integer $\geqq 3$. Let ${{𝒮}}_n$ be the set of simple spherical $n$-knots in $S^{n+2}$. Let $m$ be an integer $\geqq 4$. We prove that the map $j:{{𝒮}}_{2m}\to {{𝒮}}_{2m+4}$ 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 $P\otimes Q^{{\otimes }^k}$ denote the knot product $P\otimes \underset{k}{\underbrace{Q\otimes \cdots \otimes Q}}$. We prove the following: The $(4k+1)$-dimensional submanifold $J\otimes {\mathrm{\text{Hopf}}}^{{\otimes }^k}$$\subset S^{4k+3}$ is obtained from $K\otimes {\mathrm{\text{Hopf}}}^{{\otimes }^k}$ 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 this paper for the definitions of all local-moves in this abstract.

We prove the following: Let $a,b,a^{\prime },b^{\prime }$, and $k$ be positive integers. If the $(a,b)$ torus link is pass-move-equivalent to the $(a^{\prime },b^{\prime })$ torus link, then the Brieskorn manifolds, $\mathrm{\Sigma }(a,b,\underset{2k}{\underbrace{2,\dots ,2}})$ and $\mathrm{\Sigma }(a^{\prime },b^{\prime },\underset{2k}{\underbrace{2,\dots ,2}})$, are diffeomorphic 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 $\ge 2$. We prove the following: The $(4k+2)$-submanifold $J\otimes {\mathrm{\text{Hopf}}}^{{\otimes }^k}$$\subset S^{4k+4}$ is obtained from $K\otimes {\mathrm{\text{Hopf}}}^{{\otimes }^k}$ by a single $(2k+1,2k+2)$-pass-move, which is a local-move on $(4k+2)$-dimensional submanifolds contained in $S^{4k+4}$.