Triangulations of hyperbolic 3-manifolds admitting strict angle structures

TL;DR

This paper constructs topological ideal triangulations with strict angle structures for cusped hyperbolic 3-manifolds under certain homology conditions, advancing understanding of geometric decompositions and their limitations.

Contribution

It introduces a method to produce ideal triangulations with strict angle structures for a broad class of hyperbolic 3-manifolds, including knot complements, under homology assumptions.

Findings

Constructed triangulations with strict angle structures under homology conditions

Provided examples of triangulations lacking strict angle structures due to homology obstructions

Showed that the produced triangulations are not always geometric

Abstract

It is conjectured that every cusped hyperbolic 3-manifold has a decomposition into positive volume ideal hyperbolic tetrahedra (a "geometric" triangulation of the manifold). Under a mild homology assumption on the manifold we construct topological ideal triangulations which admit a strict angle structure, which is a necessary condition for the triangulation to be geometric. In particular, every knot or link complement in the 3-sphere has such a triangulation. We also give an example of a triangulation without a strict angle structure, where the obstruction is related to the homology hypothesis, and an example illustrating that the triangulations produced using our methods are not generally geometric.