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 ${\Delta }_{K_{+}}-{\Delta }_{K_{+}}=(t-1)\cdot {\Delta }_{K_0}$, where ${\Delta }_K$ 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 ${\Delta }_{K_{+}}-{\Delta }_{K_{+}}=(t-1)\cdot {\Delta }_{K_0}$, where ${\Delta }_K$ is a polynomial to represent the Alexander polynomial of $K$. We show another relation where $K_{+}$, $K_{-}$, and $K_0$ satisfy the formula $\mathrm{Arf}{K}_{+}-\mathrm{Arf}{K}_{-}=\{|b{P}_{4k+2}\cap I(K_0)|+1\}mod2,$ 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.

9pages, 8figures