Constructing Infinitely Many Geometric Triangulations Of The Figure Eight Knot Complement

TL;DR

This paper demonstrates the existence of infinitely many geometric ideal triangulations of the figure eight knot complement, marking the first such construction for a cusped hyperbolic 3-manifold, with limitations noted for related manifolds.

Contribution

It provides the first explicit construction of infinitely many geometric ideal triangulations for a cusped hyperbolic 3-manifold, specifically the figure eight knot complement.

Findings

Infinitely many geometric ideal triangulations of the figure eight knot complement are constructed.

The approach does not extend to the figure eight sister manifold.

It remains unknown whether similar triangulations exist for the sister manifold.

Abstract

This paper considers "geometric" ideal triangulations of cusped hyperbolic 3-manifolds, i.e. decompositions into positive volume ideal hyperbolic tetrahedra. We exhibit infinitely many geometric ideal triangulations of the figure eight knot complement. As far as we know, this is the first construction of infinitely many geometric triangulations of a cusped hyperbolic 3-manifold. In contrast, our approach does not extend to the figure eight sister manifold, and it is unknown if there are infinitely many geometric triangulations for this manifold.