Knot Energies and Knot Invariants

To record what has happened, ancient people tie knots. | I ching, the Chinese classic of 1027{771 B.C. Knots are fascinating objects. When fastening a rope, the distinction between a knot and a \slip-knot" (one that can be undone by pulling) must have been recognized very early in human history. We even developed a subconscious about knots: When we are puzzled or troubled, we have a feeling of being knotted somewhere. However, the mathematical study of knots started much later. It was inspired in the middle of the nineteenth century by the vortex theory of uid dynamics (see 11] for a vivid description of this history). The development of modern topology in the rst half of the twentieth century provided a solid background for a mathematical theory of knots. Yet we only began to see the full scope of knot theory in the last decade, starting with the discovery of the Jones polynomial in 1984 (see 14] for a survey of the history of knot theory up to Jones' discovery). In 1989, Witten generalized the Jones polynomial using his Chern-Simons path integral. Finally, in 1990-92, the development of knot theory culminated in the theory of Vassiliev knot invariants, which provides probably the most general framework for the study of the combinatorics of knots. Through the study of Vassiliev knot invariants, we see that although the abundance of knots in varieties is distinctively visible, this abundance does not come from any randomness. The combinatorics of knots embraces almost all fundamental symmetries of mathematics and physics that we know. Such a pervasive nature is not common among topological and geometric objects that mathematicians favor. For the reader's convenience, we have collected several excellent expository papers on these developments in the references (see 1, 2, 4, 8, 15]). Geometers are restless in their eeort to search for geometric objects with \max-imal homogeneity". Here, of course, the measurement of homogeneity are diierent in diierent situations. Actually, it is the key point to recognize in a given geometric setting what should be the measurement of homogeneity. Thus, in classical Rie-mannian geometry, we know that various curvatures are the key measurement of homogeneity; we measure length or area for immersions of circles and surfaces into 1