# The profinite completions of knot groups determine the Alexander polynomials

## TL;DR

This paper proves that the profinite completions of knot groups uniquely determine their Alexander polynomials, linking algebraic properties of knot groups to classical knot invariants.

## Contribution

It establishes that isomorphic profinite completions of knot groups imply identical Alexander polynomials, a novel connection between group completions and knot invariants.

## Key findings

- Profinite completions of knot groups determine Alexander polynomials.
- Isomorphic profinite groups imply identical Alexander polynomials.
- Links algebraic group properties to classical knot invariants.

## Abstract

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 $\Delta_J(t)$ and $\Delta_K(t)$ coincide.