Algebraic invariants, mutation, and commensurability of link complements

Details Algebraic invariants, mutation, and commensurability of link complements Algebraic invariants, mutation, and commensurability of link complements

2012-02-03

arxiv.org/abs/1202.0765v2

Mathematics

Geometric Topology

48 pages, 12 figures. Corrected typos, added acknowledgments and a reference in Remark 1 of intro

Scientific paper

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 Q(i,\sqrt{2}), but some have integral traces while others do not.

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