Verified computations for closed hyperbolic 3‐manifolds. Issue 2 (19th December 2020)

• Record Type: Journal Article
• Authors: Goerner, Matthias
• Is Part Of: Bulletin of the London Mathematical Society. Volume 53:Issue 2(2021)
• Page Start: 596
• Page End: 618
• Publication Date: 2020-12-19
• Subjects: 57M50 -- 65G40 (primary); Mathematics -- Periodicals; 510
• DOI: 10.1112/blms.12445
• Languages: English
• ISSNs: 0024-6093

Abstract

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 on 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.