The profinite completions of knot groups determine the Alexander polynomials

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