Ribbon-moves of 2-knots: the torsion linking pairing and the $\tilde{\eta }$-invariants of 2-knots

Abstract

We discuss the ribbon-move for 2-knots, which is a local move. Let $K$ and $K^{\prime }$ be 2-knots. Then we have: Suppose that $K$ and $K^{\prime }$ are ribbon-move equivalent. (1) Let $\mathrm{Tor}{H}_1({\tilde{X}}_K;\mathbb{Z})$ (resp. $\mathrm{Tor}{H}_1({\tilde{X}}_{K^{\prime }};\mathbb{Z})$) be the $\mathbb{Z}$-torsion submodule of the Alexander module $H_1({\tilde{X}}_K;\mathbb{Z})$ (resp. $H_1({\tilde{X}}_{K^{\prime }};\mathbb{Z})$). Then $\mathrm{Tor}{H}_1({\tilde{X}}_K;\mathbb{Z})$ is isomorphic to $\mathrm{Tor}{H}_1({\tilde{X}}_{K^{\prime }};\mathbb{Z})$ not only as $\mathbb{Z}$-modules but also as $\mathbb{Z}[t,t^{-1}]$-modules. (2) The Farber-Levine pairing for $K$ is equivalent to that for $K^{\prime }$. (3) The set of the values of the $\Q/\Z$-valued $\tilde{\eta }$ invariants for $K$ is equivalent to that for $K^{\prime }$.

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