The profinite completions of knot groups determine the Alexander polynomials

摘要

We study several properties of the completed group ring $widehat{Z}[mskip-2mu[t^{widehat{Z}}]mskip-2mu]$ and the completed Alexander modules of knots. Then we prove that if the profinite completions of the groups of two knots $J$ and $K$ are isomorphic, then their Alexander polynomials $Delta_J(t)$ and $Delta_K(t)$ coincide.