86:. A fundamental property of the O'Hara energy function is that infinite energy barriers exist for passing the knot through itself. With some additional restrictions, O'Hara showed there were only finitely many knot types with energies less than a given bound. Later, Freedman, He, and Wang removed these restrictions.
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to an ideal configuration that minimizes the electrostatic energy. Naively defined, the integral for the energy will diverge and a regularization trick from physics, subtracting off a term from the energy, is necessary. In addition the knot could change knot type under evolution unless
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on the space of all knot conformations. A conformation of a knot is a particular embedding of a circle into three-dimensional space. Depending on the needs of the energy function, the space of conformations is restricted to a sufficiently
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An electrostatic energy of polygonal knots was studied by
Fukuhara in 1987 and shortly after a different, geometric energy was studied by Sakuma. In 1988,
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functions. A property of the functional often requires that evolution of the knot under
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states that two electric charges of the same sign will repel each other as the
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231:; He, Zheng-Xu; Wang, Zhenghan (1994), "Möbius energy of knots and unknots",
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Langevin, R.; O'Hara, J. (2005), "Conformally invariant energies of knots",
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The collection of problems on “Low dimensional topology and related matters”
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The most common type of knot energy comes from the intuition of the knot as
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Sakuma, M. (1987), "Problem no. 8", in Kojima, S.; Negami, S. (eds.),
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class. For example, one may consider only polygonal circles or
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defined a knot energy based on electrostatic energy,
148:Journal of the Institute of Mathematics of Jussieu
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107:, Academic Press, Boston, MA, pp. 443–451,
103:Fukuhara, Shinji (1988), "Energy of a knot",
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191:O'Hara, Jun (1991), "Energy of a knot",
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59:inverse square of the distance
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41:does not change knot type.
206:10.1016/0040-9383(91)90010-2
134:Langevin & O'Hara (2005)
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170:10.1017/S1474748005000058
130:(in Japanese), p. 7
234:Annals of Mathematics
229:Freedman, Michael H.
51:electrically charged
18:physical knot theory
105:A fête of topology
68:self-intersections
63:electric potential
237:, Second Series,
45:Electrical charge
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39:gradient descent
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161:math.GT/0409396
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84:Möbius energy
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55:Coulomb's law
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275:Knot theory
241:(1): 1–50,
22:knot energy
90:References
80:Jun O'Hara
74:Variations
26:functional
269:Category
193:Topology
255:1259363
215:1098918
178:2135138
113:0928412
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156:arXiv
24:is a
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243:doi
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174:MR
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109:MR
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