#### Volume 18, issue 5 (2018)

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The profinite completions of knot groups determine the Alexander polynomials

### Jun Ueki

Algebraic & Geometric Topology 18 (2018) 3013–3030
##### Abstract

We study several properties of the completed group ring $\stackrel{̂}{ℤ}\left[\left[{t}^{\stackrel{̂}{ℤ}}\right]\right]$ 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 ${\Delta }_{J}\left(t\right)$ and ${\Delta }_{K}\left(t\right)$ coincide.

##### Keywords
profinite completion, profinite group ring, knot, branched covering
##### Mathematical Subject Classification 2010
Primary: 57M27
Secondary: 20E18, 20E26, 57M12