Abstract
What length of rope (of given diameter) is required to tie a particular knot ? Or, to turn the problem around, given an embedded curve, how thick a regular neighborhood of the curve also is embedded ? Intuitively, the diameter of the possible rope is bounded by the distance between strands at the closest crossing in the knot. But of course the distance between two points along a curve goes to zero as the points approach each other, so to make the notion precise, we need to exclude some neighborhood of the diagonal.
The first author was funded in part by NSF grant DMS 94-04278, and both authors enjoyed the hospitality of the IMA during the Spring 1996 term.
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References
S.S. Chern, Curves and surfaces in Euclidean space, In S.S. Chern, editor, Studies in Global Geometry and Analysis, 16–56, Math. Assoc. Amer., 1967.
Yuanan Diao, Claus Ernst, and E.J. Janse Van Rensburg, Energies of knots, Preprint, 1996.
Michael H. Freedman, Zheng-Xu He, and Zhenghan Wang On the Möbius energy of knots and unknots, Annals of Math. 139 (1994), 1–50.
Mikhael Gromov, Homotopical effects of dilatation, J. Differential Geometry 13 (1978), 303–310.
Mikhael Gromov, Filling Riemannian manifolds, J. Differential Geometry 18 (1983), 1–147.
Mikhael Gromov, J. Lafontaine, and P. Pansu, Structures métriques pour les variétés riemanniennes,Cedic/Fernand Nathan, Paris, 1981.
E.J. Janse Van Rensburg, Personal communication, June 1996.
Vsevolod Katritch, Jan Bednar, Didier Michoud, Robert G. Scharein, Jacques Dubochet, and Andrzej Stasiak, Geometry and physics of knots, Nature 384 (November 1996), 142–145.
Robert B. Kusner and John M. Sullivan, Möbius energies for knots and links, surfaces and submanifolds, In Wiliam H Kazez, editor, Geometric Topology, 570–604, Amer. Math. Soc./International Press, 1997. Proceedings of the Georgia Int’l Topology Conference, August 1993.
Richard A. Litherland, Jon Simon, Oguz Durumeric, and Eric Raw-Don, Thickness of knots, Preprint, 1996.
Jun O’hara, Energy of a knot, Topology 30 (1991), 241–247.
Jun O’hara, Family of energy functionals of knots, Topology Appl. 48 (1992), 147–161.
Jun O’hara, Energy functionals of knots II, Topology Appl. 56 (1994), 45–61.
Andrzej Stasiak, Lecture at the AMS special session on Physical Knot Theory, Iowa City, March 1996.
John M. Sullivan, Lecture at the IMA workshop on Topology and Geometry in Polymer Science, June 1996.
Tatiana Toro, Lecture at the IMA workshop on Topics Related to Nonlinear PDE, March 1996.
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Kusnert, R.B., Sullivant, J.M. (1998). On Distortion and Thickness of Knots* . In: Whittington, S.G., De Sumners, W., Lodge, T. (eds) Topology and Geometry in Polymer Science. The IMA Volumes in Mathematics and its Applications, vol 103. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1712-1_7
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DOI: https://doi.org/10.1007/978-1-4612-1712-1_7
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