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.

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