The complement of the figure-eight knot geometrically bounds

TL;DR

This paper demonstrates that the complement of the figure-eight knot can be realized as a boundary of a finite volume hyperbolic 4-manifold, providing the first such example and the smallest known volume among them.

Contribution

It proves that the figure-eight knot complement geometrically bounds a hyperbolic 4-manifold, introducing the first example of such a bounding hyperbolic knot complement.

Findings

The figure-eight knot complement geometrically bounds a hyperbolic 4-manifold.

It is the smallest volume example among known geometrically bounding hyperbolic manifolds.

Some hyperbolic 3-manifolds tessellated by ideal tetrahedra embed in hyperbolic 4-manifolds.

Abstract

We show that some hyperbolic 3-manifolds which are tessellated by copies of the regular ideal hyperbolic tetrahedron embed geodesically in a complete, finite volume, hyperbolic 4-manifold. This allows us to prove that the complement of the figure-eight knot geometrically bounds a complete, finite volume hyperbolic 4-manifold. This the first example of geometrically bounding hyperbolic knot complement and, amongst known examples of geometrically bounding manifolds, the one with the smallest volume.