In 1992, Jun O'Hara [O'H] introduced the family of knot energies \[ E^{\alpha,p}(\gamma) = \iint\limits_{\mathbb{R}/\mathbb{Z}\times\mathbb{R}/\mathbb{Z}} \left(\frac1{|\gamma(x)-\gamma(y)|^\alpha}-\frac1{{d_\gamma(x,y)}^\alpha}\right)^p |\gamma'(x)||\gamma'(y)|\;\mathrm{d}x\;\mathrm{d}y \]

The range of O'Hara's energies

where \(\gamma:\mathbb{R}/\mathbb{Z}\to\mathbb{R}^3\) is an absolutely continuous embedded curve and \(d_\gamma(x,y)\) denotes the intrinsic distance of \(\gamma(x)\) and \(\gamma(y)\). They are well-defined with values in \([0,\infty]\), non-singular if \((\alpha-2)p<1\), and self-avoiding if \(\alpha p\ge2\).

The Möbius energy

The element \((\alpha,p)=(2,1)\) was named Möbius energy by Freedman, He, and Wang [FHW] after they discovered its invariance under Möbius transformations in \(\mathbb{R}^3\). Exploiting this particular property, they have been able to prove \(C^{1,1}\) regularity of local minimizers. Subsequently He [H] noticed that the first derivative of the energy has a pseudo-differential structure which allowed for proving \(C^\infty\) regularity.

The characterization of the energy space of \(E^{2,1}\), namely the fractional Sobolev space \(W^{3/2,2}\), by Blatt [Bl] allowed for extending He's result to critical points by employing techniques from the theory of fractional harmonic maps into spheres [BRS]. The proof does not rely on Möbius invariance at all.

In the context of the Smale Conjecture, it is a meaningful question whether there exist nontrivial critical points of \(E^{2,1}\) within the unknot class [Bu].

What about other energies?

He's arguments can be applied to the energies \(E^{\alpha,1}\), \(\alpha\in(2,3)\), as well [R]. This is essentially due to the fact that all energy spaces are Hilbert spaces [Bl]. Exploiting the latter result, one can prove \(C^\infty\) smoothness of stationary points without imposing any assumption on their initial regularity [BR1].

The situation of \(p>1\) is much more involved and subject of ongoing research.

Interestingly, despite modeling different geometrical concepts, other energy families such as the tangent-point potential or the integer Menger curvature turn out to be very similar to O'Hara's energies from an analyst's perspective [BR2, BR3]. An overview is provided in [BR4].

References

[Bl] Simon Blatt. Boundedness and regularizing effects of O’Hara’s knot energies. J. Knot Theory Ramifications, 21(1):1250010, 2012. [ doi ]
[Bu] Ryan Budney. A gorgeous but incomplete proof of “The Smale Conjecture”. Oct. 2016. [ blog ]
[BR1] Simon Blatt and Philipp Reiter. Stationary points of O’Hara’s knot energies. Manuscripta Math., 40(1-2):29–50, 2013. [ arXiv | doi ]
[BR2] Simon Blatt and Philipp Reiter. Regularity theory for tangent-point energies: The non-degenerate sub-critical case. Adv. Calc. Var., 8(2):93–116, 2015. [ arXiv | doi ]
[BR3] Simon Blatt and Philipp Reiter. Towards a regularity theory for integral Menger curvature. Ann. Acad. Sci. Fenn. Math., 40(1):149–181, 2015. [ arXiv | doi ]
[BR4] Simon Blatt and Philipp Reiter. Modeling repulsive forces on fibres via knot energies. Mol. Based Math. Biol., 2(1):56–72, 2014. [ preprint | doi ]
[BRS] Simon Blatt, Philipp Reiter, and Armin Schikorra. Harmonic analysis meets critical knots. Critical points of the Möbius energy are smooth. Trans. Amer. Math. Soc., 368:6391–6438, 2016. [ arXiv | doi ]
[FHW] Michael H. Freedman, Zheng-Xu He, and Zhenghan Wang. Möbius energy of knots and unknots. Ann. of Math. (2), 139(1):1–50, 1994. [ jstor ]
[H] Zheng-Xu He. The Euler-Lagrange equation and heat flow for the Möbius energy. Comm. Pure Appl. Math., 53(4):399–431, 2000.[ doi ]
[O'H] Jun O’Hara. Family of energy functionals of knots. Topology Appl., 48(2):147–161, 1992. [ open archive ]
[R] Philipp Reiter. Repulsive knot energies and pseudodifferential calculus for O’Hara’s knot energy family E(α), α∈[2,3). Math. Nachr., 285(7):889–913, 2012. [ preprint | doi ]