Two notes on the O'Hara energies

Shoya Kawakami

doi:10.3934/dcdss.2020384

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2021, Volume 14, Issue 3: 953-970. Doi: 10.3934/dcdss.2020384

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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

Early access: June 2020

Published: March 2021

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Abstract

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.

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Mathematics Subject Classification: Primary: 46E35, 49M25; Secondary: 57K10.

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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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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 $
	Number of vertices $ n $	$ 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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References

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