The profinite completions of knot groups determine the Alexander polynomials

Abstract

We study several properties of the completed group ring $\hat{\mathbb{Z}}[[t^{\hat{\mathbb{Z}}}]]$ 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.

14 pages

References in corpus (7)

• Ricci flow with surgery on three-manifolds
• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• Profinite detection of 3-manifold decompositions
• Profinite rigidity and surface bundles over the circle
• Some 3-manifold groups with the same finite quotients
• $p$-adic Mahler measure and $\mathbb{Z}$-covers of links
• Torsion homology growth beyond asymptotics

Cited by in corpus (10)

• Profinite detection of 3-manifold decompositions
• Finite-volume hyperbolic 3-manifolds are almost determined by their finite quotient groups
• $p$-adic Mahler measure and $\mathbb{Z}$-covers of links
• Profinite rigidity for twisted Alexander polynomials
• Torsion homology growth beyond asymptotics
• The Iwasawa invariants of ${\mathbb{Z}}_p^d$-covers of links
• The $p$-adic limits of class numbers in ${\mathbb{Z}}_p$-towers
• Profinite rigidity in the SnapPea census
• On profinite rigidity amongst free-by-cyclic groups I: the generic case
• Alexander polynomial, Dijkgraaf-Witten invariant, and Seifert fibred surgery