eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-06-08
25:1
25:17
10.4230/LIPIcs.SoCG.2020.25
article
The Next 350 Million Knots
Burton, Benjamin A.
1
The University of Queensland, Brisbane, Australia
The tabulation of all prime knots up to a given number of crossings was one of the founding problems of knot theory in the 1800s, and continues to be of interest today. Here we extend the tables from 16 to 19 crossings, with a total of 352 152 252 distinct non-trivial prime knots.
The tabulation has two major stages: (1) a combinatorial enumeration stage, which involves generating a provably sufficient set of candidate knot diagrams; and (2) a computational topology stage, which involves identifying and removing duplicate knots, and certifying that all knots that remain are topologically distinct. In this paper we describe the many different algorithmic components in this process, which draw on graph theory, hyperbolic geometry, knot polynomials, normal surface theory, and computational algebra. We also discuss the algorithm engineering challenges in solving difficult topological problems systematically and reliably on hundreds of millions of inputs, despite the fact that no reliably fast algorithms for these problems are known.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol164-socg2020/LIPIcs.SoCG.2020.25/LIPIcs.SoCG.2020.25.pdf
Computational topology
knots
3-manifolds
implementation