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Knots connected by wide ribbons
Journal article   Peer reviewed

Knots connected by wide ribbons

Susan C Brooks, Oguz Durumeric and Jonathan Simon
Journal of knot theory and its ramifications, Vol.28(12), p.1950071
10/01/2019
DOI: 10.1142/S0218216519500718
url
https://arxiv.org/pdf/1808.00154View
Open Access

Abstract

A ribbon is a smooth mapping (possibly self-intersecting) of an annulus S-1 x I in 3-space having constant width R. Given a regular parametrization x(s), and a smooth unit vector field u(s) based along x, for a knot K, we may define a ribbon of width R associated to x and u as the set of all points x(s) + ru(s), r is an element of [0, R]. For large R, ribbons, and their outer edge curves, may have self-intersections. In this paper, we analyze how the knot type of the outer ribbon edge x(s) + Ru(s) relates to that of the original knot K. Generically, as R -> infinity, there is an eventual constant knot type. This eventual knot type is one of only finitely many possibilities which depend just on the vector field u. The particular knot type within the finite set depends on the parametrized curves x(s), u(s), and their interactions. We demonstrate a way to control the curves and their parametrizations so that given two knot types K-1 and K-2, we can find a smooth ribbon of constant width connecting curves of these two knot types.
 Mathematics Physical Sciences Science & Technology

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