The profinite completions of knot groups determine the Alexander polynomials

Ueki, Jun

doi:10.2140/agt.2018.18.3013

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The profinite completions of knot groups determine the Alexander polynomials

Jun Ueki

Algebraic & Geometric Topology 18 (2018) 3013–3030

DOI: 10.2140/agt.2018.18.3013

Abstract

We study several properties of the completed group ring $\hat{ℤ} [[t^{\hat{ℤ}}]]$ 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 $Δ_{J} (t)$ and $Δ_{K} (t)$ coincide.

Keywords

profinite completion, profinite group ring, knot, branched covering

Mathematical Subject Classification 2010

Primary: 57M27

Secondary: 20E18, 20E26, 57M12

References

Publication

Received: 24 September 2017

Revised: 21 February 2018

Accepted: 5 March 2018

Published: 22 August 2018

Authors

Jun Ueki
Department of Mathematics School of System Design and Technology Tokyo Denki University Tokyo Japan

Volume 18, issue 5 (2018)