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Möbius invariance of knot energy. (English) Zbl 0776.57003

The energy of a rectifiable curve \(\gamma: (\mathbb{R}\) or \(S^ 1)\to\mathbb{R}^ 3\) is given by \[ E(\gamma)=\iint\left\{{1\over |\gamma(u)- \gamma(v)|^ 2}-{1\over D(\gamma(u),\gamma(v))^ 2}\right\}|\dot\gamma(u)| |\dot\gamma(v)| du dv, \] where \(D(\gamma(u),\gamma(v))\) is the shortest arc distance between \(\gamma(u)\) and \(\gamma(v)\) on the curve. This energy is invariant under Möbius transformations. This fact leads to many interesting conclusions, for example: among all rectifiable loops, round circles have the least energy (4) and any loop with energy 4 is a round circle; any smooth prime knot is equivalent to one of minimal energy; any rectifiable loop with energy less than \(6\pi+4\) is unknotted.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
49Q10 Optimization of shapes other than minimal surfaces
53A04 Curves in Euclidean and related spaces
57N45 Flatness and tameness of topological manifolds
58E30 Variational principles in infinite-dimensional spaces
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References:

[1] K. Ahara, Energy of a knot, screened at Topology Conf., Univ. of Hawaii, August 1990, K. H. Dovermann, organizer.
[2] M. H. Freedman and Z.-X. He, On the ’energy’ of knots and unknots (to appear).
[3] Jun O’Hara, Energy of a knot, Topology 30 (1991), no. 2, 241 – 247. · Zbl 0733.57005
[4] Jun O’Hara, Family of energy functionals of knots, Topology Appl. 48 (1992), no. 2, 147 – 161. · Zbl 0769.57006
[5] Jun O’Hara, Energy functionals of knots, Topology Hawaii (Honolulu, HI, 1990) World Sci. Publ., River Edge, NJ, 1992, pp. 201 – 214. · Zbl 1039.58500
[6] C. Ernst and D. W. Sumners, The growth of the number of prime knots, Math. Proc. Cambridge Philos. Soc. 102 (1987), no. 2, 303 – 315. · Zbl 0632.57007
[7] W. T. Tutte, A census of planar maps, Canad. J. Math. 15 (1963), 249 – 271. · Zbl 0115.17305
[8] D. J. A. Welsh, On the number of knots, preprint. · Zbl 0799.57001
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