Self-contact Sets for 50 Tightly Knotted and Linked Tubes

TL;DR

This paper introduces a new numerical method to compute self-contact sets in tightly knotted tubes, providing improved bounds on ropelength for various knots and links, advancing understanding of their geometric properties.

Contribution

The authors develop a constrained gradient-descent approach to compute self-contact sets, improving upon previous simulated annealing methods for multiple knot types.

Findings

New self-contact sets for 50 knots and links

Improved upper bounds on ropelength for knots with 9 or fewer crossings

Enhanced numerical techniques for knot geometry analysis

Abstract

We report on new numerical computations of the set of self-contacts in tightly knotted tubes of uniform circular cross-section. Such contact sets have been obtained before for the trefoil and figure eight knots by simulated annealing -- we use constrained gradient-descent to provide new self-contact sets for those and 48 other knot and link types. The minimum length of all unit diameter tubes in a given knot or link type is called the ropelength of that class of curves. Our computations yield improved upper bounds for the ropelength of all knots and links with 9 or fewer crossings except the trefoil.