American Institute of Mathematical Sciences

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March  2021, 14(3): 953-970. doi: 10.3934/dcdss.2020384

Two notes on the O'Hara energies

Shoya Kawakami

Graduate School of Science and Engineering, Saitama University, Shimo-Okubo 255, Sakura-ku, Saitama, Japan

Received  January 2019 Revised  February 2020 Published  June 2020

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.

Keywords: Knot energy, Sobolev-Slobodeckij space, discrete approximation, minimizer, numerical computation.
Mathematics Subject Classification: Primary: 46E35, 49M25; Secondary: 57K10.
Citation: Shoya Kawakami. Two notes on the O'Hara energies. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 953-970. doi: 10.3934/dcdss.2020384
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. O'Hara, Family of energy functionals of knots, Topology Appl., 48 (1992), 147-161.  doi: 10.1016/0166-8641(92)90023-S.  Google Scholar

[12]

J. O'Hara, Energy functionals of knots. Ⅱ, Topology Appl., 56 (1994), 45-61.  doi: 10.1016/0166-8641(94)90108-2.  Google Scholar

[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.  Google Scholar

[14]

S. Scholtes, Discrete Möbius energy, J. Knot Theory Ramifications, 23 (2014), 1450045, 16 pp. doi: 10.1142/S021821651450045X.  Google Scholar

[15]

J. K. Simon, Energy functions for polygonal knots, J. Knot Theory Ramifications, 3 (1994), 299-320.  doi: 10.1142/S021821659400023X.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. O'Hara, Family of energy functionals of knots, Topology Appl., 48 (1992), 147-161.  doi: 10.1016/0166-8641(92)90023-S.  Google Scholar

[12]

J. O'Hara, Energy functionals of knots. Ⅱ, Topology Appl., 56 (1994), 45-61.  doi: 10.1016/0166-8641(94)90108-2.  Google Scholar

[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.  Google Scholar

[14]

S. Scholtes, Discrete Möbius energy, J. Knot Theory Ramifications, 23 (2014), 1450045, 16 pp. doi: 10.1142/S021821651450045X.  Google Scholar

[15]

J. K. Simon, Energy functions for polygonal knots, J. Knot Theory Ramifications, 3 (1994), 299-320.  doi: 10.1142/S021821659400023X.  Google Scholar

Figure 1.  Graphs of $ e_\alpha (n) $ (The vertical and horizontal axes show values of $ e_\alpha(n) $ and numbers of vertices $ n = 2^k $ ($ k = 2,3, \cdots , 20 $), respectively)
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Figure 2.  Values of $ \mathcal{E}^{2,30}_{2^k}( {\boldsymbol{g}}_{2^k}) $
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Figure 3.  Values of $ \mathcal{E}^{2,30}_n( {\boldsymbol{g}}_n) $ when $ n \leq 100 $ (Round points and diamond points show values when $ n $ is even and odd, respectively)
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Table 1.  Examples of $ \Phi $ (Ranges of $ \alpha $)
$ \Phi(x) = x^\alpha $ $ \Phi(x) = x^\alpha\log (x+1) $ $ \Phi(x) = 1-e^{-x^\alpha}+x^{2\alpha} /2 $ ($ x \in [0,C_{\text b} \mathcal{L}] $)
Theorem 2.2 $ [2/p , \infty) $ $ [2/p-1 , \infty) $ $ [1/p, \infty) $
Theorem 2.3 $ (1/p , 2+1/p) $ $ (1/p , 1/p+1) $ $ (1/p , 2+1/p) $
Remark 2 $ [2/p , 2+1/p) $ $ [2/p,1/p+1) $ $ [2/p , 2+1/p) $ $ (p >1) $
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$ \Phi(x) = x^\alpha $ $ \Phi(x) = x^\alpha\log (x+1) $ $ \Phi(x) = 1-e^{-x^\alpha}+x^{2\alpha} /2 $ ($ x \in [0,C_{\text b} \mathcal{L}] $)
Theorem 2.2 $ [2/p , \infty) $ $ [2/p-1 , \infty) $ $ [1/p, \infty) $
Theorem 2.3 $ (1/p , 2+1/p) $ $ (1/p , 1/p+1) $ $ (1/p , 2+1/p) $
Remark 2 $ [2/p , 2+1/p) $ $ [2/p,1/p+1) $ $ [2/p , 2+1/p) $ $ (p >1) $
Table 2.  Numerical calculation of $ \mathcal{L}( {\boldsymbol{f}}_0)^{\alpha -2}\mathcal{E}^{\alpha , 1}( {\boldsymbol{f}}_0) $ when $ 2 \leq \alpha < 3 $ (D: Values of discretization, D$ / $A: Divisions of value of discretization when $ n = 4194304 $ by analytic value)
Number of vertices $ n $ $ \alpha $
$ 2 $ $ 2.1 $ $ 2.3 $ $ 2.5 $ $ 2.7 $ $ 2.9 $
D $ 4 $ $ 1 $ $ 1.147365 $ $ 1.500936 $ $ 1.949372 $ $ 2.516555 $ $ 3.232177 $
$ 8 $ $ 2.325253 $ $ 2.739102 $ $ 3.780728 $ $ 5.187945 $ $ 7.085586 $ $ 9.640817 $
$ 16 $ $ 3.134412 $ $ 3.754475 $ $ 5.372714 $ $ 7.672833 $ $ 10.95137 $ $ 15.64031 $
$ 32 $ $ 3.562332 $ $ 4.320470 $ $ 6.363289 $ $ 9.408493 $ $ 13.99728 $ $ 20.99456 $
$ 64 $ $ 3.780229 $ $ 4.626457 $ $ 6.969742 $ $ 10.61781 $ $ 16.42130 $ $ 25.87401 $
$ 128 $ $ 3.889916 $ $ 4.790718 $ $ 7.341313 $ $ 11.46626 $ $ 18.37252 $ $ 30.38526 $
$ 256 $ $ 3.944913 $ $ 4.878765 $ $ 7.569466 $ $ 12.06415 $ $ 19.95194 $ $ 34.58121 $
$ 512 $ $ 3.972446 $ $ 4.925946 $ $ 7.709746 $ $ 12.48634 $ $ 21.23325 $ $ 38.49223 $
$ 1024 $ $ 3.986220 $ $ 4.951228 $ $ 7.796054 $ $ 12.78472 $ $ 22.27356 $ $ 42.14019 $
$ 2048 $ $ 3.993109 $ $ 4.964776 $ $ 7.849171 $ $ 12.99567 $ $ 23.11844 $ $ 45.54354 $
$ 4096 $ $ 3.996555 $ $ 4.972036 $ $ 7.881865 $ $ 13.14482 $ $ 23.80467 $ $ 48.71889 $
$ 8192 $ $ 3.998277 $ $ 4.975926 $ $ 7.901990 $ $ 13.25028 $ $ 24.36205 $ $ 51.68157 $
$ 16384 $ $ 3.999139 $ $ 4.978011 $ $ 7.914378 $ $ 13.32485 $ $ 24.81478 $ $ 54.44584 $
$ 32768 $ $ 3.999569 $ $ 4.979129 $ $ 7.922004 $ $ 13.37758 $ $ 25.18251 $ $ 57.02499 $
$ 65536 $ $ 3.999785 $ $ 4.979727 $ $ 7.926698 $ $ 13.41487 $ $ 25.48120 $ $ 59.43143 $
$ 131072 $ $ 3.999892 $ $ 4.980048 $ $ 7.929588 $ $ 13.44124 $ $ 25.72381 $ $ 61.67671 $
$ 262144 $ $ 3.999946 $ $ 4.980220 $ $ 7.931366 $ $ 13.45988 $ $ 25.92087 $ $ 63.77161 $
$ 524288 $ $ 3.999973 $ $ 4.980312 $ $ 7.932461 $ $ 13.47306 $ $ 26.08094 $ $ 65.72639 $
$ 1048576 $ $ 3.999987 $ $ 4.980362 $ $ 7.933135 $ $ 13.48238 $ $ 26.21093 $ $ 67.55013 $
$ 2097152 $ $ 4.000004 $ $ 4.980401 $ $ 7.933568 $ $ 13.48900 $ $ 26.31651 $ $ 69.25143 $
$ 4194304 $ $ 3.999997 $ $ 4.980402 $ $ 7.933807 $ $ 13.49362 $ $ 26.40257 $ $ 70.84417 $
Analytic values $ 4 $ $ 4.980419 $ $ 7.934215 $ $ 13.50489 $ $ 26.77342 $ $ 92.95965 $
D$ / $A $ 0.999999 $ $ 0.999997 $ $ 0.999949 $ $ 0.999166 $ $ 0.986148 $ $ 0.762096 $
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Number of vertices $ n $ $ \alpha $
$ 2 $ $ 2.1 $ $ 2.3 $ $ 2.5 $ $ 2.7 $ $ 2.9 $
D $ 4 $ $ 1 $ $ 1.147365 $ $ 1.500936 $ $ 1.949372 $ $ 2.516555 $ $ 3.232177 $
$ 8 $ $ 2.325253 $ $ 2.739102 $ $ 3.780728 $ $ 5.187945 $ $ 7.085586 $ $ 9.640817 $
$ 16 $ $ 3.134412 $ $ 3.754475 $ $ 5.372714 $ $ 7.672833 $ $ 10.95137 $ $ 15.64031 $
$ 32 $ $ 3.562332 $ $ 4.320470 $ $ 6.363289 $ $ 9.408493 $ $ 13.99728 $ $ 20.99456 $
$ 64 $ $ 3.780229 $ $ 4.626457 $ $ 6.969742 $ $ 10.61781 $ $ 16.42130 $ $ 25.87401 $
$ 128 $ $ 3.889916 $ $ 4.790718 $ $ 7.341313 $ $ 11.46626 $ $ 18.37252 $ $ 30.38526 $
$ 256 $ $ 3.944913 $ $ 4.878765 $ $ 7.569466 $ $ 12.06415 $ $ 19.95194 $ $ 34.58121 $
$ 512 $ $ 3.972446 $ $ 4.925946 $ $ 7.709746 $ $ 12.48634 $ $ 21.23325 $ $ 38.49223 $
$ 1024 $ $ 3.986220 $ $ 4.951228 $ $ 7.796054 $ $ 12.78472 $ $ 22.27356 $ $ 42.14019 $
$ 2048 $ $ 3.993109 $ $ 4.964776 $ $ 7.849171 $ $ 12.99567 $ $ 23.11844 $ $ 45.54354 $
$ 4096 $ $ 3.996555 $ $ 4.972036 $ $ 7.881865 $ $ 13.14482 $ $ 23.80467 $ $ 48.71889 $
$ 8192 $ $ 3.998277 $ $ 4.975926 $ $ 7.901990 $ $ 13.25028 $ $ 24.36205 $ $ 51.68157 $
$ 16384 $ $ 3.999139 $ $ 4.978011 $ $ 7.914378 $ $ 13.32485 $ $ 24.81478 $ $ 54.44584 $
$ 32768 $ $ 3.999569 $ $ 4.979129 $ $ 7.922004 $ $ 13.37758 $ $ 25.18251 $ $ 57.02499 $
$ 65536 $ $ 3.999785 $ $ 4.979727 $ $ 7.926698 $ $ 13.41487 $ $ 25.48120 $ $ 59.43143 $
$ 131072 $ $ 3.999892 $ $ 4.980048 $ $ 7.929588 $ $ 13.44124 $ $ 25.72381 $ $ 61.67671 $
$ 262144 $ $ 3.999946 $ $ 4.980220 $ $ 7.931366 $ $ 13.45988 $ $ 25.92087 $ $ 63.77161 $
$ 524288 $ $ 3.999973 $ $ 4.980312 $ $ 7.932461 $ $ 13.47306 $ $ 26.08094 $ $ 65.72639 $
$ 1048576 $ $ 3.999987 $ $ 4.980362 $ $ 7.933135 $ $ 13.48238 $ $ 26.21093 $ $ 67.55013 $
$ 2097152 $ $ 4.000004 $ $ 4.980401 $ $ 7.933568 $ $ 13.48900 $ $ 26.31651 $ $ 69.25143 $
$ 4194304 $ $ 3.999997 $ $ 4.980402 $ $ 7.933807 $ $ 13.49362 $ $ 26.40257 $ $ 70.84417 $
Analytic values $ 4 $ $ 4.980419 $ $ 7.934215 $ $ 13.50489 $ $ 26.77342 $ $ 92.95965 $
D$ / $A $ 0.999999 $ $ 0.999997 $ $ 0.999949 $ $ 0.999166 $ $ 0.986148 $ $ 0.762096 $
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