Up: Geometry Forum Articles

Evelyn Sander
geometry.college, June 17, 1994.

Rob Kusner of the University of Massachusetts and John Sullivan of the University of Minnesota have been at the Geometry Center for the last month to work on energy minimization in the study of knots and links. Basically, minimal energy configurations are those for which parts of the knot or link stay far apart. For example, an unknotted loop has as its minimal energy a round circle. They hope that looking at energies will give a way to distinguish between different knots and links.

A particularly nice class of links is the class Hopf links. These are links composed of particular great circles on the three-sphere which come from the Hopf fibration, defined as follows. The three-sphere is embedded in four-space. Regarding four-space as complex two-space, Hopf circles are the intersection of the sphere and the complex linear subspaces (two-dimensional real subspaces). Equivalently, they are points on the three-sphere such that the ratio of the two complex coordinates is constant. Hopf circles turn out to be great circles of the three-sphere. In fact, each point on the three-sphere is contained in exactly one Hopf circle. This division into Hopf circles is called the Hopf fibration. Any two Hopf circles link exactly once. (Their linking number is +1, giving the circles the same orientation as the complex linear subspaces in which they lie.) Thus a collection of Hopf circles form a link.

For Hopf links, the associated energy depends only on the distance between loops in four-space, since each loop in the link is itself a round circle. The energy Kusner and Sullivan use is proportional to one over the square of the distance. However, it is not necessary to work with the three-sphere in four-space to compute this quantity; for each Hopf circle, there is a uniquely defined point on the two-sphere. This is the point on the Riemann sphere given by the constant ratio of the two complex coordinates for points on a Hopf circle. Minimizing the associated energy for configurations of points on the two-sphere is equivalent to looking at the original energy function for Hopf links on the three-sphere.

The associated energy function for points on the two-sphere turns out to be the same as the classical Coulomb's law energy between charged particles, in other words, inversely proportional to the distances between points in three-space. Some classical physics papers discuss the energy minimization problem of point charges on a sphere. However, work on the problem stopped around 1912 due to the discovery of quantum mechanics.

Here are the first four particle minimal energy configurations; for two particles, antipodal points have minimal energy. Assuming that they are at the north and south poles, the corresponding link consists of loops which are the intersections of the three-sphere with the (z,0)- and the (0,w)-complex lines. For three points, the minimum energy occurs when the three are evenly spaced on a great circle of the sphere. Four points have minimum energy when placed in a regular tetrahedron, but there is another critical configuration; can you find it? The minimum energy configurations are known for up to six particles. For larger numbers of points, little is known about critical configurations and the number of stable configurations (local minima). Last summer Kusner and Sullivan discovered, using Ken Brakke's program Surface Evolver, that there are two different stable configurations of sixteen particles. One appears to be the global minimum; the other has slightly more energy. Kusner and Sullivan are currently using Morse theory to try to classify all the critical configurations of charges on the sphere up through sixteen particles.