# Verified computations for closed hyperbolic 3-manifolds

## TL;DR

This paper presents a verified computational method for confirming hyperbolic structures on closed 3-manifolds using interval arithmetic, extending previous techniques and solving an open problem in knot theory.

## Contribution

It introduces a new theoretical result ensuring redundancy among edge equations, enabling verification of hyperbolic structures on closed 3-manifolds.

## Key findings

- Successfully verified structures on known manifolds
- Determined hyperbolic branched double covers for knots up to 14 crossings
- Extended verification methods to closed 3-manifolds

## Abstract

Extending methods first used by Casson, we show how to verify a hyperbolic structure on a finite triangulation of a closed 3-manifold using interval arithmetic methods. A key ingredient is a new theoretical result (akin to a theorem by Neumann-Zagier and Moser for ideal triangulations upon which HIKMOT is based) showing that there is a redundancy among the edge equations if the edges avoid "gimbal lock". We successfully test the algorithm on known examples such as the orientable closed manifolds in the Hodgson-Weeks census and the bundle census by Bell. We also tackle a previously unsolved problem and determine all knots and links with up to 14 crossings that have a hyperbolic branched double cover.