Algebraic invariants, mutation, and commensurability of link complements

TL;DR

This paper constructs hyperbolic link complements using tangle gluings, explores their commensurability relations, and shows how mutation affects their geometric and arithmetic properties, including volume, trace fields, and cusp parameters.

Contribution

It introduces a method to generate hyperbolic link complements with controlled properties and analyzes how mutation influences their geometric and arithmetic invariants.

Findings

Different volume link complements are incommensurable.

Mutation can produce large families of manifolds with identical volume and scissors class.

Some manifolds have integral traces, others do not.

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