Hidden Symmetries and Commensurability of 2-Bridge Link Complements

TL;DR

This paper proves that non-arithmetic hyperbolic 2-bridge link complements lack hidden symmetries, characterizes their commensurability classes, and explores implications for 3-manifold topology using tiling analysis.

Contribution

It establishes the absence of hidden symmetries in non-arithmetic hyperbolic 2-bridge link complements and classifies their commensurability with only two known examples.

Findings

Non-arithmetic hyperbolic 2-bridge link complements have no hidden symmetries.

The only commensurable hyperbolic 2-bridge link complements are the figure-eight knot and the 6_2^2 link.

Hyperbolic 2-bridge link complements cannot irregularly cover other hyperbolic 3-manifolds.

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.