Abstract
Energy functions on knots are continuous and scale-invariant functions defined from knot conformations into non-negative real numbers. The infimum of an energy function is an invariant which defines (not necessarily unique) “canonical conformations” of knots in three space. Many (infinite) hierarchies of energy functions for knots in the mathematical and physical science literature have been studied, each energy function with its own (characteristic) set of properties. In this paper we examine energy functions, and classify them as either basic, strong, charge or tight,depending on the properties of the energies for different knot conformations. Knot invariants derived from these energy functions are expected to be useful.
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References
G. Buck, Applications of the Projection Energy for Knots, to appear.
G. Buck and J. Orloff, Computing canonical conformations for knots, Top. Appl., 51 (1993), 246–253.
Y. Diao, C. Ernst and Janse Van. Rensburg, Thicknesses of Knots, Preprint.
Y. Diao, C. Ernst and Janse Van. Rensburg, In search of a good polygonal knot energy, to appear in J. Knot Theo. Ram..
M.H. Freedman, Z-X He and Z. Wang Möbius Energy of Knots and Unknots, Ann. Math., 139 (1994), 1–50.
S. FukuharaEnergy of a knot, in The Fête of Topology, Academic Press (1987), 443–451.
M. GromovHomotopical Effects of Dilation, J. Diff. Geom., 13 (1978), 303–310.
M. GromovFilling Riemannian Manifolds, J. Diff. Geom., 18 (1983), 1–147.
V. Katritch, J. Bednar, D. Michoud, R.G. Scharein, J. Dubochet and A. StasiakGeometry and physics of knots, Nature, 384 (1996) 142–145.
R.B. Kusner and J.M. Sullivan, Möbius Energies for Knots and Links, Surfaces and Submanifolds, MSRI Preprint, No. 026–94.
J. Langer and D.A. Singer, Curve Straightening and a Minimax Argument for Closed Elastic Curves, Topology, 24 (1985), 75–88.
R.A. Litherland, J.K. Simon, O. Durumer1c and E. Rawdon, Thickness of Knots, Preprint.
J. MilnorOn the Total Curvature of Knots, Ann. Math., 52 (1950), 248–257.
Moffatt, The Energy Spectrum of Knots and Links, Nature, 347 (1990), 367–369.
J. O’haraEnergy of a Knot, Topology, 30 (1991), 241–247.
J. O’hara, Family of Energy Functionals of Knots, Top. App., 48 (1992), 147–161.
J. O’hara, Energy Functionals of Knots II, Top. App., 56 (1994), 45–61.
J.K. SimonEnergy Functions for Polygonal Knots, J. Knot Theo. Ram., 3, No. 3 (1994), 299–320.
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Diao, Y., Ernst, C., van Rensburg, E.J.J. (1998). Properties of Knot Energies. 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_5
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DOI: https://doi.org/10.1007/978-1-4612-1712-1_5
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