The profinite completions of knot groups determine the Alexander polynomials

Abstract

We study several properties of the completed group ring $\hat{\mathbb{Z}}\left[\left[t^{\hat{\mathbb{Z}}}\right]\right]$ 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 ${\mathrm{\Delta }}_J\left(t\right)$ and ${\mathrm{\Delta }}_K\left(t\right)$ coincide.