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

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].

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.