文章基本信息

标题：The Next 350 Million Knots
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作者：Benjamin A. Burton
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2020
卷号：164
页码：25:1-25:17
DOI：10.4230/LIPIcs.SoCG.2020.25
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：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.
关键词：Computational topology; knots; 3-manifolds; implementation