A Family of M Obius Invariant 2-knot Energies

After reviewing energy functionals for 1-dimensional knots and links, we deene a family of MM obius invariant energy functionals E s for surfaces embedded in R n. These functionals are all nite for smoothly embedded compact surfaces and innnite for self-intersecting immersed surfaces. They treat disconnected surfaces and connected sums of surfaces correctly. For suuciently negative s, E s is not bounded from below. For suuciently positive s, the evidence to date suggests that E s is bounded from below, but we have not yet found a proof. We also discuss alternate methods of deening surface energy. Several years ago Jun O'Hara O1, O2, O3] deened a functional for embeddings of circles in R n that he called the energy of a knot. Bryson, Freedman, He and Wang BFHW, FHW] proved that this energy is MM obius invariant, and bounded the complexity of knots by their energy. Since then others, notably Doyle and Schramm D,S], have modiied and reformulated O'Hara's energy. The formulations are all essentially equivalent, but each exhibits a diierent important property of the energy. In this paper we generalize some of these ideas to 2 dimensions and deene a family of energy functionals for embedded surfaces. These functionals are diierent from, and postdate, the surface energy deened by Rob Kusner and John Sullivan, which is discussed elsewhere in these proceedings KS]. Section 1 is a review of the 1-dimensional energy. We begin with the unregularized energy formula and discuss its properties. We then list several ideas for regularization and show how they are essentially equivalent. Each of these methods suggests a regularization in 2 dimensions, most of which turn out not to be equivalent. The material in this section is not original, but rather reeects past work by many authors O1, O2, O3, BFHW, FHW, D, S]. In section 2 we generalize O'Hara's original regularization to 2 dimensions. There is some freedom in the choice of regularization, yielding a 1-parameter family of MM obius invariant surface energies E s that diier by multiples of the Willmore functional W]. We compute the energies of several examples, and establish that these energy functionals have all but one of the desirable properties shared by the 1-dimensional energy. The one uncertain property is boundedness, which is discussed further in section 4.