Energy and Length of Knots

Knots are idealized 1-dimensional loops that tangle themselves in 3-space. They have been studied, for more than 100 years, primarily as abstract mathematical objects even though the original interest in the subject seems to be based in physics. There is now interest in reinvesting the mathematical abstractions with physical-like properties such as thickness L] Si] DEJ] St] LSDR] or self-repelling energy Fu] O1-O4] BS] BO1-2] Si2] Lo] DEJ]. The motivation is partly chemistry/biology and partly the lure of the mathematics itself. By modeling knots with physical properties, new invariants of knots can be deened and there is hope for better understanding of how knotted and tangled laments (simple loops, links of several loops, or tangled spatial graphs) behave in real systems such as DNA gel electrophoresis DSKC] DC] DC2] Ketc] SSC] This paper considers and relates several notions of energy and other measures of geometric complexity for knots. Some of the results described here were announced in B] and Si4]. The discussion of Theorem 4 given here is a summary; the detailed proof is presented in BS2]. We may hope to show that various energies are related to each other, e.g. by inequalities saying that one energy is less than some function of another, and that they also are related to intuitive geometric measures of knot complexity such as compaction (a long knot contained in a small ball) or average crossing number. Another idea, thickness (or, rather, its reciprocal, the rope-length of a knot) may be viewed either as an energy or as a naive geometric measure of complexity; in any case, it also may be related to the others by various inequalities. The general pattern of the theorems is that knots which seem complicated according to one measure, also must be complicated according to others. This leads us to believe that while the various energies etc. are deened diierently (and are diierent), they all are capturing, at least approximately, the same intuitive idea of one knot being more complicated than another. One theorem common to the various energy functions is that there are only nitely many knot types that can be realized by knots below a given energy level, and that all knots below some level are unknotted. There also are interesting questions about existence, uniqueness (and/or rigidity) of minimum-energy conformations. The situation is clearer for polygonal knots (minima The authors are supported in part by NSF Grant #DMS9407132 …