On knot complements that decompose into regular ideal dodecahedra

TL;DR

This paper proves that only two specific knot complements can be decomposed into regular ideal dodecahedra, confirming a conjecture about their uniqueness in hyperbolic 3-manifold decompositions.

Contribution

It establishes the uniqueness of knot complements decomposing into regular ideal dodecahedra, solving part of a conjecture by Neumann and Reid.

Findings

Only two knot complements decompose into two regular ideal dodecahedra.

These two are the only such knot complements, confirming their uniqueness.

Provides a partial proof of a conjecture regarding hyperbolic 3-manifold decompositions.

Abstract

Aitchison and Rubinstein constructed two knot complements that can be decomposed into two regular ideal dodecahedra. This paper shows that these knot complements are the only knot complements that decompose into n regular ideal dodecahedra, providing a partial solution to a conjecture of Neumann and Reid.