Algebraic invariants, mutation, and commensurability of link complements

Eric Chesebro and Jason DeBlois

Vol. 267 (2014), No. 2, 341–398

DOI: 10.2140/pjm.2014.267.341

Abstract

We construct a family of hyperbolic link complements by gluing tangles along totally geodesic four-punctured spheres, then 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 nonisometric manifolds with the same volume and scissors congruence class. Depending on the choice of mutation, these manifolds may be commensurable or incommensurable, distinguished in the latter case by cusp parameters. All have trace field $\mathbb{Q}\left(i,\sqrt{2}@\right)$; some have integral traces while others do not.

Keywords

commensurability, mutation, Bloch invariant, link, trace field

Mathematical Subject Classification 2010

Primary: 57M10, 57M50

Secondary: 20F55, 22E40

Milestones

Received: 18 March 2012

Revised: 13 August 2013

Accepted: 3 September 2013

Published: 11 May 2014