Is there an optimal way to tie a knot in space, or to embed a more general submanifold? And is there a natural way to evolve any knot to an optimal equivalent one, so that we could detect whether two knots are really the same?
One approach to such questions is to associate to any knot (or more general submanifold) an energy, and look for minimizers or critical points of this energy. If the energy is infinite for immersions which are not embeddings, then presumably its gradient flow will prevent self-crossings and preserve knot type. One way to get an energy with such an infinite barrier against self-crossings is to think of spreading charge along the knot and then consider the electrostatic potential. Such an energy for knots was introduced by Ohara and studied by Freedman, He and Wang. We define an analogous knot energy for k-dimensional submanifolds in n-dimensional euclidean space Rn (or the sphere Sn). Our knot energy is again a repulsive potential between points on the submanifold, depending only on first-order data. It is given by a regularized inverse power law, with the power chosen to make the energy scale-invariant and the regularization to make it invariant under conformal (Möbius) transformations of the ambient space.
The gradient flow of our knot energy appears to lead to optimal embeddings, both theoretically and computationally. In particular, for classical knots and links, we have used our knot energy to create an effective algorithm, implemented in Brakke's evolver, to untangle complicated curves to a simple representative for their knot type by gradient descent. In most cases, we reach the energy minimum. For instance, all unknots we have tried evolve to the round circle, and both curves in the famous Perko pair evolve to the same configuration, proving they are the same knot. Thus in most cases, this is an effective algorithm for classifying knots. However, we have also found certain links with several distinct local minima at different energy values; for these rare cases, gradient descent methods will not always reach the same final configuration.
Knot energies for curves were introduced into mathematics motivated by physical considerations; they are closely related to classically defined energies for divergence-free vector fields which arise in modeling incompressible fluid flow. These new knot energies may help to model certain natural phenomena. For example, the inverse power laws in knot energies seem related to some of the energies involved in arising in protein folding problems. And recent experiments suggest that the speed of DNA knots in electrophoresis gels is correlated to other notions of knot energy.
For surfaces, we have previously modeled another Möbius-invariant energy, the elastic bending energy popularized by Willmore, in the evolver. It is known that this energy describes the behavior of lipid vesicles, and in fact such vesicles have been observed undergoing Möobius transformations in laboratory experiments. To model these vesicles in more detail, one might like to include a van der Waals interaction between different surface molecules; perhaps our Möbius-invariant knot energy would be an interesting choice for modeling such a nonlocal interaction. Our knot energies in higher dimensions or codimensions do not have obvious physical interpretation or application, although they have been useful, for example, in the topological study of knotted spheres in four-space.
In this paper, we define our family of knot energies for submanifolds of arbitrary dimensions, and then explore the particular case of energies for knots and links, and alternative regularizations there. We then show why we should not expect minimizers for composite knots. We discuss the discretizations we have implemented in the evolver, and their success in untangling complicated unknots. We relate the knot energy to other measures of geometric complexity, like crossing number and ropelength. Next, we discuss critical points for the energy which are guaranteed by symmetry, and the construction in this way of Hopf links with distinct local minima for the energy. Finally, we have computed energy minimizers for all knots and links up through eight crossings, and present the results of this computation and a table of their energies; our appendix shows stereoscopic pictures of the Möbius-energy minimizers.