• Previous Article
    A doubly splitting scheme for the Caginalp system with singular potentials and dynamic boundary conditions
  • DCDS-S Home
  • This Issue
  • Next Article
    Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative
doi: 10.3934/dcdss.2020384

Two notes on the O'Hara energies

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.

Citation: Shoya Kawakami. Two notes on the O'Hara energies. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020384
References:
[1]

A. AbramsJ. CantarellaJ. H. G. FuM. 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. FreedmanZ.-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. AbramsJ. CantarellaJ. H. G. FuM. 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. FreedmanZ.-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)
Figure 2.  Values of $ \mathcal{E}^{2,30}_{2^k}( {\boldsymbol{g}}_{2^k}) $
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)
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) $
$ \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 $
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 $
[1]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463

[2]

Manuel Friedrich, Martin Kružík, Jan Valdman. Numerical approximation of von Kármán viscoelastic plates. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 299-319. doi: 10.3934/dcdss.2020322

[3]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[4]

Christopher S. Goodrich, Benjamin Lyons, Mihaela T. Velcsov. Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound. Communications on Pure & Applied Analysis, 2021, 20 (1) : 339-358. doi: 10.3934/cpaa.2020269

[5]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

[6]

Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255

[7]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[8]

Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250

[9]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[10]

Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems by stages. Journal of Geometric Mechanics, 2020, 12 (4) : 607-639. doi: 10.3934/jgm.2020029

[11]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268

[12]

Kai Yang. Scattering of the focusing energy-critical NLS with inverse square potential in the radial case. Communications on Pure & Applied Analysis, 2021, 20 (1) : 77-99. doi: 10.3934/cpaa.2020258

[13]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046

[14]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339

[15]

Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078

[16]

Vieri Benci, Marco Cococcioni. The algorithmic numbers in non-archimedean numerical computing environments. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020449

[17]

Reza Lotfi, Zahra Yadegari, Seyed Hossein Hosseini, Amir Hossein Khameneh, Erfan Babaee Tirkolaee, Gerhard-Wilhelm Weber. A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020158

[18]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

[19]

Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213

[20]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (42)
  • HTML views (188)
  • Cited by (0)

Other articles
by authors

[Back to Top]