Mutations and short geodesics in hyperbolic 3-manifolds

TL;DR

This paper constructs large classes of hyperbolic knot complements with identical volume and length spectra, analyzes their uniqueness within commensurability classes, and explores conditions under which mutations preserve geometric properties.

Contribution

It provides explicit constructions of incommensurable hyperbolic knot complements sharing volume and length spectra, and establishes conditions for mutations to preserve these spectra.

Findings

Constructed large classes of incommensurable hyperbolic knot complements with same volume and length spectrum.

Proved these knot complements are unique within their commensurability classes based on cusp shape analysis.

Identified geometric and topological conditions under which mutations preserve the initial length spectrum.

Abstract

In this paper, we explicitly construct large classes of incommensurable hyperbolic knot complements with the same volume and the same initial (complex) length spectrum. Furthermore, we show that these knot complements are the only knot complements in their respective commensurabiltiy classes by analyzing their cusp shapes. The knot complements in each class differ by a topological cut-and-paste operation known as mutation. Ruberman has shown that mutations of hyperelliptic surfaces inside hyperbolic 3-manifolds preserve volume. Here, we provide geometric and topological conditions under which such mutations also preserve the initial (complex) length spectrum. This work requires us to analyze when least area surfaces could intersect short geodesics in a hyperbolic 3-manifold.