Verified Computations for Hyperbolic 3-Manifolds

Abstract

For a given cusped 3-manifold M admitting an ideal triangulation, we describe a method to rigorously prove that either M or a filling of M admits a complete hyperbolic structure via verified computer calculations. Central to our method is an implementation of interval arithmetic and Krawczyk’s test. These techniques represent an improvement over existing algorithms as they are faster while accounting for error accumulation in a more direct and user-friendly way.

2000 AMS Subject Classification::

• 57M50
• 65G40

Keywords:

• hyperbolic 3-manifold
• interval arithmetic
• Krawczyk’s test
• verified numerical computations

Notes

²One needs a rational arithmetic scheme to compute the rank of a matrix with integer entries rigorously. In general, such a scheme is more expensive computationally than a floating-point scheme.