Script for Knot Energies:

Final version, May 1998

Knot theory tries to detect when knots are equivalent. But it is hard to tell how to simplify a complicated knot like this.

One approach is to associate an energy to every closed curve in space, and evolve the curve to reduce this energy. For instance, if we spread electric charge along the curve, minimizing the potential energy would make the strands repel each other.

We use a renormalized double integral over pairs of points on the curve, of one over distance squared. The computer models this experimentally with polygonal curves.

Since our starting curve was an unknot, though quite tangled, it evolves to the round circle, which has the minimum possible energy for any curve.

The simplest truly knotted knot is a trefoil or overhand knot. Minimizing its energy, we get a symmetric curve, also constructible theoretically. Rewinding, we can watch the evolution again, to see how three-dimensional the curve becomes. The minimizing configuration is a (2,3)-curve on the surface of a round torus.

Our knot energy is Möbius-invariant, so as we take a tour through the conformal group, all of these are equivalent energy-minimizing trefoils. As we change the conformal view, sometimes we see two-fold or three-fold Euclidean symmetry. The curve actually has a continuous Möbius symmetry: it is the orbit of a point under a rigid rotation of the three-sphere.

The (3,4)-torus knot evolves first towards a symmetric critical configuration, but this seems unstable, and eventually the symmetry gets broken. We reach the presumed minimum, with energy about one percent less. Rewinding, we can watch the evolution again.

Similarly, the (3,5)-torus knot has a symmetric configuration, on the surface of a round torus, but this seems unstable, and evolves to break its symmetry.

This connected sum of two trefoils degenerates under energy minimization. In the limit, it has has just twice the energy of a minimizing trefoil, since the two pieces hardly interact. It is known that all prime knots have minimizing configurations. Here, conformal transformations could keep one of the two trefoils from shrinking, but not both at once.

Two distinct ten-crossing diagrams, which represent the same knot, the so-called Perko pair, evolve to what seem like distinct configurations. But in fact, these curves are Möbius-equivalent.

Our energy also works for links like the Borromean rings. By Möbius-invariance, this energy-minimizing configuration has various forms, including three ovals in perpendicular planes.