Trace fields and commensurability of link complements

TL;DR

This paper explores the limitations of trace fields as invariants for classifying hyperbolic 3-manifolds, demonstrating that different manifolds can share the same trace field, thus challenging their effectiveness.

Contribution

It constructs infinite families of hyperbolic link complements and 3-manifolds with identical trace fields but distinct commensurability classes, revealing new insights into trace field invariants.

Findings

Constructed infinite families of incommensurable link complements with same trace field

Demonstrated existence of hyperbolic 3-manifolds sharing trace fields

Showed that mutant links can have nonintegral traces

Abstract

This paper investigates the strength of the trace field as a commensurability invariant of hyperbolic 3-manifolds. We construct an infinite family of two-component hyperbolic link complements which are pairwise incommensurable and have the same trace field, and infinitely many 1-cusped finite volume hyperbolic 3-manifolds with the same property. We also show that the two-component link complements above have integral traces, but each has a mutant with a nonintegral trace.