Hidden Symmetries and Commensurability of 2-Bridge Link Complements

Abstract

In this paper, we show that any non-arithmetic hyperbolic $2$-bridge link complement admits no hidden symmetries. As a corollary, we conclude that a hyperbolic $2$-bridge link complement cannot irregularly cover a hyperbolic $3$-manifold. By combining this corollary with the work of Boileau and Weidmann, we obtain a characterization of $3$-manifolds with non-trivial JSJ-decomposition and rank two fundamental groups. We also show that the only commensurable hyperbolic $2$-bridge link complements are the figure-eight knot complement and the $6_2^2$ link complement. Our work requires a careful analysis of the tilings of ${\mathbb{R}}^2$ that come from lifting the canonical triangulations of the cusps of hyperbolic $2$-bridge link complements.

This is the final (accepted) version of this paper

Cited by in corpus (2)

• Symmetries and hidden symmetries of $(\epsilon ,d_L)$-twisted knot complements
• L-spaces, taut foliations and fibered hyperbolic two-bridge links