Local move formulae for \\the Alexander polynomials of $n$-knots

Abstract

It is well-known:Suppose there are three 1-dimensional links $K_+$, $K_-$, $K_0$ such that $K_+$, $K_-$, and $K_0$ coincide out of a 3-ball $B$ trivially embedded in $S^3$ and that $K_+\cap B$, $K_-\cap B$, and $K_0\cap B$ are drawn as follows. Then $Δ_{K_+}-Δ_{K_+}=(t-1)\cdotΔ_{K_0}$, where $Δ_{K}$ is the Alexander polynomial of $K$. We know similar formulae of other invariants of 1-dimensional knots and links. (The Jones polynomial etc.) It is natural to ask: Suppose there are two $n$-dimensional knots $K_+$, $K_-$ and a submanifold $K_0$ such that $K_+$, $K_-$, and $K_0$ coincide out of a $n$-ball $B$ trivially embedded in $S^{n+2}$. Then is there a relation in $K_+\cap B$, $K_-\cap B$, and $K_0\cap B$ with the following property(*)? (*)If $K_+$, $K_-$, and $K_0$ satisfy this relation, an invariant of $K_+$, that of $K_-$, and that of $K_0$ satisfy a fixed relation. In this paper we pove there are such a relation where $K_+$, $K_-$, and $K_0$ satisfy the formula $Δ_{K_+}-Δ_{K_+}=(t-1)\cdotΔ_{K_0}$, where $Δ_{K}$ is a polynomial to represent the Alexander polynomial of $K$. We show another relation where $K_+$, $K_-$, and $K_0$ satisfy the formula ${\mathrm{Arf}}K_+-{\mathrm{Arf}}K_-=\{|bP_{4k+2}\cap I(K_0)|+1\}mod 2,$ where (1)$I()$ is the inertia group. and $I(K_0)$ is the inertia group of a smooth manifold which is orientation preserving diffeomorphic to $K_0$. (2)For a group $G$, $|G|$ denote the order of $G$. A local move formula is a relation of an invariant of a few knots related by a local move as above.