We study several properties of the completed group ring and the completed Alexander modules of knots. Then we prove that if the profinite completions of the groups of two knots and are isomorphic, then their Alexander polynomials and coincide.
"The profinite completions of knot groups determine the Alexander polynomials." Algebr. Geom. Topol. 18 (5) 3013 - 3030, 2018. https://doi.org/10.2140/agt.2018.18.3013