Commensurability classes of twist knots

Abstract

In this paper we prove that if $M_K$ is the complement of a non-fibered twist knot $K$ in ${\mathbb{S}}^3$, 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.

10 pages, 3 figures