Narrow Results

In this paper we discuss some fundamental issues regarding knot energy functions. These include the existence of minimum values of energy functions of smooth knots and energy functions of polygonal knots within a knot type, the convergence of these minimum values in the case of polygonal knot energy and the convergence of the corresponding polygons where these minimum values are attained. When the polygonal knot energy is derived from a smooth knot energy, will the minimal polygonal knot energies converge to the infimum of the smooth knot energy? Do the corresponding polygons converge to a smooth knot at which the smooth energy achieves its minimal value? We show that one cannot expect these to be true in general and outline certain conditions that would ensure a positive answer to some of the above questions.

In this paper, we construct a conformally invariant functional for two-component links called the zone modulus of the link. Its main property is to give a sufficient condition for a link to be split.

The zone modulus is a positive number, and its lower bound is 1. To construct a link with modulus arbitrarily close to 1, it is sufficient to consider two small disjoint spheres each one far from the other and then to construct a link by taking a circle enclosed in each sphere. Such a link is a split link. The situation is different when the link is non-split: we will prove that the modulus of a non-split link is greater than . This value of the modulus is realized by a special configuration of linked circles called the Clifford link.

As a corollary, we show that if the thickness of a non-split two-component link embedded in S³ is equal to , then the link is the standard geometric Hopf link.

An energy function for smooth knots is defined using the concept of thickness and proved to be a good indicator of the complexity of knots. This energy function has the advantage that it can be measured experimentally. We carry out an experiment to measure the energies for knots up to 8 crossings and compare our data with energies obtained elsewhere through numerical simulations.

Energy functions on knots are continuous and scale-invariant functions defined from knot conformations into non-negative real numbers. The infimum of an energy function is a knot invariant which defines (not necessarily unique) “canonical conformations” of knots in three space. In this paper we examine strategies of how to find these canonical conformations and compute or measure their minimal energy. Furthermore, we discuss properties that energy functions should have if one wants to compute canonical conformations of knots with minimal energy. Two types of energies are discussed in more detail, the first type consists of energies of C² or C¹ knots defined using the concept of thickness of a knot; the second type is a polygonal energy which should be well suited to numerical computation since it has all the properties discussed earlier.