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Two notes on the O'Hara energies
Graduate School of Science and Engineering, Saitama University, Shimo-Okubo 255, Sakura-ku, Saitama, Japan |
The O'Hara energies, introduced by Jun O'Hara in 1991, were proposed to answer the question of what is a "good" figure in a given knot class. A property of the O'Hara energies is that the "better" the figure of a knot is, the less the energy value is. In this article, we discuss two topics on the O'Hara energies. First, we slightly generalize the O'Hara energies and consider a characterization of its finiteness. The finiteness of the O'Hara energies was considered by Blatt in 2012 who used the Sobolev-Slobodeckij space, and naturally we consider a generalization of this space. Another fundamental problem is to understand the minimizers of the O'Hara energies. This problem has been addressed in several papers, some of them based on numerical computations. In this direction, we discuss a discretization of the O'Hara energies and give some examples of numerical computations. Particular one of the O'Hara energies, called the Möbius energy thanks to its Möbius invariance, was considered by Kim-Kusner in 1993, and Scholtes in 2014 established convergence properties. We apply their argument in general since the argument does not rely on Möbius invariance.
References:
[1] |
A. Abrams, J. Cantarella, J. H. G. Fu, M. Ghomi and R. Howard,
Circles minimize most knot energies, Topology, 42 (2003), 381-394.
doi: 10.1016/S0040-9383(02)00016-2. |
[2] |
S. Blatt, Boundedness and regularizing effects of O'Hara's knot energies, J. Knot Theory Ramifications, 21 (2012), 1250010, 9 pp.
doi: 10.1142/S0218216511009704. |
[3] |
G. Dal Maso, An Introduction to $\Gamma$-convergence, Progress in Nonlinear Diffrential Equations and their Applications, 8. Birkhäuser Boston, Boston, MA, 1993.
doi: 10.1007/978-1-4612-0327-8. |
[4] |
M. H. Freedman, Z.-X. He and Z. H. Wang,
Möbius energy of knots and unknots, Ann. of Math. (2), 139 (1994), 1-50.
doi: 10.2307/2946626. |
[5] |
A. Ishizeki and T. Nagasawa, Decomposition of generalized O'Hara's energies, (2019), arXiv: 1904.06812. Google Scholar |
[6] |
S. Kawakami, A discretization of O'Hara's knot energy and its convergence, (2019), arXiv: 1908.11172. Google Scholar |
[7] |
D. Kim and R. Kusner,
Torus knots extremizing the Möbius energy, Experiment. Math., 2 (1993), 1-9.
doi: 10.1080/10586458.1993.10504264. |
[8] |
R. Kusner and J. M. Sullivan,
Möbius-invariant knot energies, Ideal Knots, Ser. Knots Everything, World Sci. Publ., River Edge, NJ, 19 (1998), 315-352.
doi: 10.1142/9789812796073_0017. |
[9] |
S. Miyajima, Introduction to Sobolev Space and its Application, Kyoritsu Shuppan, Tokyo, 2006. Google Scholar |
[10] |
J. O'Hara,
Energy of a knot, Topology, 30 (1991), 241-247.
doi: 10.1016/0040-9383(91)90010-2. |
[11] |
J. O'Hara,
Family of energy functionals of knots, Topology Appl., 48 (1992), 147-161.
doi: 10.1016/0166-8641(92)90023-S. |
[12] |
J. O'Hara,
Energy functionals of knots. Ⅱ, Topology Appl., 56 (1994), 45-61.
doi: 10.1016/0166-8641(94)90108-2. |
[13] |
E. J. Rawdon and J. K. Simon,
Polygonal approximation and energy of smooth knots, J. Knot Theory Ramifications, 15 (2006), 429-451.
doi: 10.1142/S0218216506004543. |
[14] |
S. Scholtes, Discrete Möbius energy, J. Knot Theory Ramifications, 23 (2014), 1450045, 16 pp.
doi: 10.1142/S021821651450045X. |
[15] |
J. K. Simon,
Energy functions for polygonal knots, J. Knot Theory Ramifications, 3 (1994), 299-320.
doi: 10.1142/S021821659400023X. |
show all references
References:
[1] |
A. Abrams, J. Cantarella, J. H. G. Fu, M. Ghomi and R. Howard,
Circles minimize most knot energies, Topology, 42 (2003), 381-394.
doi: 10.1016/S0040-9383(02)00016-2. |
[2] |
S. Blatt, Boundedness and regularizing effects of O'Hara's knot energies, J. Knot Theory Ramifications, 21 (2012), 1250010, 9 pp.
doi: 10.1142/S0218216511009704. |
[3] |
G. Dal Maso, An Introduction to $\Gamma$-convergence, Progress in Nonlinear Diffrential Equations and their Applications, 8. Birkhäuser Boston, Boston, MA, 1993.
doi: 10.1007/978-1-4612-0327-8. |
[4] |
M. H. Freedman, Z.-X. He and Z. H. Wang,
Möbius energy of knots and unknots, Ann. of Math. (2), 139 (1994), 1-50.
doi: 10.2307/2946626. |
[5] |
A. Ishizeki and T. Nagasawa, Decomposition of generalized O'Hara's energies, (2019), arXiv: 1904.06812. Google Scholar |
[6] |
S. Kawakami, A discretization of O'Hara's knot energy and its convergence, (2019), arXiv: 1908.11172. Google Scholar |
[7] |
D. Kim and R. Kusner,
Torus knots extremizing the Möbius energy, Experiment. Math., 2 (1993), 1-9.
doi: 10.1080/10586458.1993.10504264. |
[8] |
R. Kusner and J. M. Sullivan,
Möbius-invariant knot energies, Ideal Knots, Ser. Knots Everything, World Sci. Publ., River Edge, NJ, 19 (1998), 315-352.
doi: 10.1142/9789812796073_0017. |
[9] |
S. Miyajima, Introduction to Sobolev Space and its Application, Kyoritsu Shuppan, Tokyo, 2006. Google Scholar |
[10] |
J. O'Hara,
Energy of a knot, Topology, 30 (1991), 241-247.
doi: 10.1016/0040-9383(91)90010-2. |
[11] |
J. O'Hara,
Family of energy functionals of knots, Topology Appl., 48 (1992), 147-161.
doi: 10.1016/0166-8641(92)90023-S. |
[12] |
J. O'Hara,
Energy functionals of knots. Ⅱ, Topology Appl., 56 (1994), 45-61.
doi: 10.1016/0166-8641(94)90108-2. |
[13] |
E. J. Rawdon and J. K. Simon,
Polygonal approximation and energy of smooth knots, J. Knot Theory Ramifications, 15 (2006), 429-451.
doi: 10.1142/S0218216506004543. |
[14] |
S. Scholtes, Discrete Möbius energy, J. Knot Theory Ramifications, 23 (2014), 1450045, 16 pp.
doi: 10.1142/S021821651450045X. |
[15] |
J. K. Simon,
Energy functions for polygonal knots, J. Knot Theory Ramifications, 3 (1994), 299-320.
doi: 10.1142/S021821659400023X. |



Theorem 2.2 | |||
Theorem 2.3 | |||
Remark 2 |
Theorem 2.2 | |||
Theorem 2.3 | |||
Remark 2 |
Number of vertices |
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D | |||||||
Analytic values | |||||||
D |
Number of vertices |
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D | |||||||
Analytic values | |||||||
D |
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