A polyhedral approach to the arithmetic and geometry of hyperbolic link complements

Abstract

We provide an elementary polyhedral approach to study and deduce results about the arithmeticity and commensurability of an infinite family of hyperbolic link complements [Formula: see text] for [Formula: see text]. The manifold [Formula: see text] is the complement of [Formula: see text] by the ([Formula: see text])-link chain [Formula: see text] and has [Formula: see text] cusps. We show that [Formula: see text] is closely related to a hyperbolic Coxeter orbifold that is commensurable to an orbifold with a single cusp. Vinberg’s arithmeticity criterion and certain cusp density and volume computations allow us to reproduce some of the main results in [20] and [18] about [Formula: see text] in a comparatively elementary and direct way. As a by-product, we give a rigorous proof of Thurston’s volume formula for [Formula: see text] and deduce that, for [Formula: see text], the volume of [Formula: see text] is strictly bigger than the volume of the ([Formula: see text])-cyclic cover over one component of the Whitehead link.