Energies of knot diagrams
A knot energy is a functional usually defined on knots embedded in 3 space into the positive real numbers. The goal is to find a (unique) minimizer of the knot energy for a given knot type and the value of this knot energy minimizer then becomes a knot invariant. Usually knot energies do not allow deformations of the embedded knot that change the knot type by increasing the energy value to infinity for curves that have self-intersections. There are many known examples of knot energies however usually only numeric approximations of the minimizer for a given knot type are possible. In this article the authors define a family of energy functionals of diagrams (instead of curves in 3 space). Here they consider a flat knot of length \(2\pi\) and cross section radius \(\epsilon\), where \(0<\epsilon<<1\), that is constrained between two parallel planes a small distance from each other. Thus the gradient descent towards a minimizer is invariant with respect to Reidemeister II and III moves but not with respect of the Reidemeister I moves. The energy functionals defined in this paper are the sum of two terms. The first called the uniformization term is defined as an integral of an arbitrary function \(f\) on the geodesic curvature. This term does not contain any crossing information of the diagram and thus does not respect the knot type. The second term called the resistance term makes use of the crossing information and is defined as a sum of the reciprocals of the areas of disks bounded by the knot diagram, where the sum is taken over all cycles in the knot diagram bounding a disk such that on in their consecutive intersections overpasses and underpasses alternate. For the exact definition the reader should look at the actual article. The authors study several versions of both the uniformization functional, where they obtain neat normal forms corresponding to minima of the functional for the special case \(f(x)=x^2\); and the resistance functional where they attempt to reduce the number of cycles that need to be considered to reduce computational complexity.
scientific article; zbMATH DE number 6097091