[1] A. Abrams, J. Cantarella, J. H. G. Fu, M. Ghomi, and R. Howard, Circles
minimize most knot energies, Topology, 42 (2) (2003), 381–394.
[2] S. Blatt, The gradient flow of the M¨
obius energy near local minimizers,
Calc. Var. Partial Differential Equations, 43 (3–4) (2012), 403–439.
[3] S. Blatt, Boundedness and regularizing effects of O’Hara’s knot energies,
J. Knot Theory Ramifications, 21 (1) (2012), 1250010, 9pp.
[4] S. Blatt, The gradient flow of O’Hara’s knot energies, Math. Ann., 370
(3–4) (2018), 993–1061.
[5] S. Blatt, A. Ishizeki, and T. Nagasawa, A M¨
obius invariant discretization
of O’Hara’s M¨
obius energy, arXiv:1809.07984.
[6] S. Blatt, A. Ishizeki, and T. Nagasawa, A M¨
obius invariant discretization
and decomposition of the M¨
obius energy, arXiv:1904.06818.
[7] S. Blatt and P. Reiter, Stationary points of O’Hara’s knot energies,
Manuscripta Math., 140 (1–2) (2013), 29–50.
60
[8] S. Blatt, P. Reiter, and A. Schikorra, Harmonic analysis meets critical
knots. Critical points of the M¨
obius energy are smooth, Trans. Amer. Math.
Soc., 368 (9) (2016), 6391–6438.
[9] S. Blatt, P. Reiter, and A. Schikorra, On O’Hara knot energies I: regularity
for critical knots, arXiv: 1905.06064.
[10] S. Blatt and N. Vorderobermeier, On the analyticity of critical points of the
M¨
obius energy, Calc. Var. Partial Differential Equations, 58 (1) (2019), 28
pp.
[11] J.-L. Brylinski, The beta function of a knot, Int. J. Math., 10 (4) (1999),
415–423.
[12] G. Dal Maso, “An introduction to Γ-convergence”, Progress in Nonlinear
Diffrential Equations and their Applications, vol. 8, Birkh¨auser Boston,
Boston, MA, 1993.
[13] M. H. Freedman, Z.-X. He, and Z. Wang, M¨
obius energy of knots and
unknots, Ann. of Math., 139 (1994), 1–50.
[14] L. G´abor, On the mean length of the chords of a closed curve, Israel J.
Math., 4 (1966), 23–32.
[15] Z.-X. He, The Euler-Lagrange equation and heat flow for the M¨
obius energy,
Comm. Pure Appl. Math., 53 (4) (2000), 399–431.
[16] A. Ishizeki and T. Nagasawa, A decomposition theorem of the M¨
obius energy
I: Decomposition and M¨
obius invariance, Kodai. Math. J., 37 (3) (2014),
737–754.
[17] A. Ishizeki and T. Nagasawa, A decomposition theorem of the M¨
obius energy
II: Variational formulae and estimates, Math. Ann., 363 (1–2) (2015), 617–
635.
[18] A. Ishizeki and T. Nagasawa, The invariance of decomposed M¨
obius energies under the inversions with center on curves, J. Knot Theory Ramifications 26 (2016), 1650009, 22 pp.
[19] A. Ishizeki and T. Nagasawa, Decomposition of generalized O’Hara’s energies, arXiv:1904.06812.
[20] S. Kawakami, A discretization of O’Hara’s knot energy and its convergence,
arXiv:1908.11172.
[21] S. Kawakami, Recent topics on the O’Hara energies, arXiv:1908.11671.
[22] S. Kawakami and T. Nagasawa, Variational formulae and estimates of
O’Hara’s knot energies, arXiv:1908.11677.
[23] D. Kim and R. Kusner, Torus knots extremizing the M¨
obius energy, Experiment. Math., 2 (1) (1993), 1–9.
[24] R. Kusner and J. M. Sullivan, On distortion and thickness of knots, in
“Ideal Knots” (Ed.: A. Stasiak, V. Katrich, L. H. Kauffman), World Scientific, Singapore, 1998, pp. 315–352.
61
[25] S. Miyajima, “Introduction to Sobolev Space and its Application”, Kyoritsu
Shuppan, Tokyo, 2006, in Japanese.
[26] J. O’Hara, Energy of a knot, Topology, 30 (2) (1991), 241–247.
[27] J. O’Hara, Family of energy functionals of knots, Topology Appl., 48 (2)
(1992), 147–161.
[28] J. O’Hara, Energy functionals of knots II, Topology Appl., 56 (1) (1994),
45–61.
[29] J. Okamoto,
Random
arXiv:1905.06657.
discretization
of
O’Hara
knot
energy,
[30] E. J. Rawdon and J. K. Simon, Polygonal approximation and energy of
smooth knots, J. Knot Theory Ramifications, 15 (4) (2006), 429–451.
[31] P. Reiter, Repulsive knot energies and pseudodifferential calculus for
O’Hara’s knot energy family E (α) , α ∈ [2, 3), Math. Nachr., 285 (7) (2012),
889–913.
[32] S. Scholtes, Discrete M¨
obius energy, J. Knot Theory Ramifications, 23
(2014), 1450045, 16 pp.
[33] J. K. Simon, Energy functions for polygonal knots, J. Knot Theory Ramifications, 3 (3) (1994), 299–320.
[34] N. Vorderobermeier, On the regularity of critical points for O’Hara’s knot
energies: From smoothness to analyticity, arXiv:1904.13129.
62
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