✦ LIBER ✦

Thickness and crossing number of knots

✍ Scribed by Gregory Buck; Jonathan Simon

Publisher: Elsevier Science
Year: 1999
Tongue: English
Weight: 836 KB
Volume: 91
Category: Article
ISSN: 0166-8641
DOI: 10.1016/s0166-8641(97)00211-3

No coin nor oath required. For personal study only.

✦ Synopsis

Given a "short" piece of rope, one can tie only "simple" knots. We make this precise by modeling "rope" as a solid tube of constant radius about a smooth core. The complexity of a knot is captured by its average crossing number which in turn bounds the minimum crossing number for the knot type. Then the ratio, L, of rope-length to radius provides an upper bound for the crossing number. Our bound is in terms of L4/", which we believe is the lowest exponent possible.

Our route for connecting rope-length of a knot to its thickness is via self-repelling knot energies, the normal energy EN(K) and the symmetric energy Es(K).

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