The University of Montana
Department of Mathematical Sciences

Technical report #23/2011

ALGEBRAIC INVARIANTS, MUTATION, AND COMMENSURABILITY
OF LINK COMPLEMENTS


ERIC CHESEBRO AND JASON DEBLOIS

Abstract

We construct a family of hyperbolic link complements, all with trace field Q(i,Sqrt(2)), by gluing tangles along totally geodesic four-punctured spheres, and investigate the commensurability relation among its members. Those with different volume are incommensurable, distinguished by their scissors congruence classes. Mutation produces arbitrarily large finite subfamilies of non-isometric manifolds with the same volume and scissors commensurability class. Depending on the choice of mutation, these manifolds may be commensurable or incommensurable, distinguished in the latter case by cusp parameters. Examples with integral and nonintegral traces are also produced.

Keywords: hyperbolic 3-manifold, commensurable, trace field, mutant

AMS Subject Classification: 57M50

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