Narrow Results

In this paper we prove that if M_K is the complement of a non-fibered twist knot K in , then M_K is not commensurable to a fibered knot complement in a ℤ/2ℤ-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 Calegari and Dunfield is satisfied for these varieties. In addition, we partially extend our results to a second infinite family of 2-bridge knots.

For an iterated function system (IFS) with contractive similarities, the post-critically finite (p.c.f.) condition and H-condition are two important separation conditions in the literature. But their relationship is not clear. In this paper, we show that if the similarity matrices of the IFS are commensurable, then the p.c.f. condition implies H-condition.