Commensurability classes of twist knots

2003/11/30 by Jim Hoste, Patrick D. Shanahan
Mathematics · #math.GT #msc:57M25

paper · pdf

published as J. Knot Theory and its Ramifications 14.1 (2005) 91-100 · 10 pages, 3 figures

arxiv created 2004/02/27 · arxiv updated 2009/12/01

Abstract

In this paper we prove that if M_K is the complement of a non-fibered twist knot K in \mathbb S³, then M_K is not commensurable to a fibered knot complement in a \mathbb Z/ 2 \mathbb Z-homology sphere. To prove this result we derive a recursive description of the character variety of twist knots and then prove that a commensurability criterion developed by D. Calegari and N. Dunfield is satisfied for these varieties. In addition, we partially extend our results to a second infinite family of 2-bridge knots.

Related

• Commensurability of 1-cusped hyperbolic 3-manifolds
• Twisted homology cobordism invariants of knots in aspherical manifolds
• Left-orderability, branched covers and double twist knots
• The non-commutative A-polynomial of twist knots
• Almost-concordance of knots in aspherical 3-manifolds