Knot energies

In 1992, Jun O'Hara [O'H] introduced the family of knot energies $$E^{\alpha ,p}(\gamma )=\underset{\mathbb{R}/\mathbb{Z}\times \mathbb{R}/\mathbb{Z}}{\iint }{(\frac{1}{|\gamma (x)-\gamma (y){|}^{\alpha }}-\frac{1}{{d_{\gamma }(x,y)}^{\alpha }})}^p|{\gamma }^{\prime }(x)||{\gamma }^{\prime }(y)|\mathrm{d}x\mathrm{d}y$$

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 $[0,\mathrm{\infty }]$, non-singular if $(\alpha -2)p<1$, and self-avoiding if $\alpha p\ge 2$.

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 $C^{\mathrm{\infty }}$ 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 $C^{\mathrm{\infty }}$ 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