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Self-contact Sets for 50 Tightly Knotted and Linked Tubes

Ted Ashton·Jason Cantarella·Michael Piatek·Eric Rawdon
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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.

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Ted Ashton, Jason Cantarella, Michael Piatek, Eric Rawdon. 2005-08-15. Self-contact Sets for 50 Tightly Knotted and Linked Tubes. https://arxiv.org/abs/math/0508248

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