On knot complements that decompose into regular ideal dodecahedra

Aitchison and Rubinstein showed there are two distinct ways to identify the faces of a pair of regular ideal dodecahedra and obtain a knot complement. This paper shows that these knot complements are the only knot complements that decompose into regular ideal dodecahedra, providing a partial solution to a conjecture of Neumann and Reid. A corollary of the main theorem classifies the hyperbolic knot complements can be decomposed into regular ideal polyhedra.