Commensurability classes of twist knots

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In this paper we prove that if Mₖ is the complement of a non-fibered twist knot K in S³, then Mₖ is not commensurable to a fibered knot complement in a Z/ 2 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.