Jun O'Hara

Family of energy functionals of knots

Abstract We define energy functionals on the space of embeddings from S1 into R 3 and show the finiteness of knot types under bounded values of those functionals.

Energy of Knots and Conformal Geometry

Just like a minimal surface is modeled on the “optimal surface” of a soap film with a given boundary curve, one can ask whether we can define an “optimal knot”, a beautiful knot which represents its knot type. Energy of knots was introduced for this purpose. The basic philosophy due to Fukuhara and Sakuma independently is as follows. Suppose there is a non-conductive knotted string which is charged uniformly in a non-conductive viscous fluid. Then it might evolve itself to decrease its electrostatic energy without intersecting itself because of Coulomb’s repulsive force until it comes to a critical point of the energy. Then we might be able to define an “optimal embedding” of a knot by an energy minimizer, which is an embedding that attains the minimum energy within its isotopy class. Thus our motivational problem can be stated as:

Energy functionals of knots II

Abstract We study an energy functional of knots, e p j ( jp > 2), that is finite valued for embedded circles and takes +∞ for circles with double points. We show that for any b ϵ R there are finitely many solid tori T 1 ,…, T m such that any knot with e p j ⩽ b can be contained in some T i in a good manner. Then we can show the existence of a minimizer of e p j in each knot type.

Renormalization of potentials and generalized centers

We generalize the Riesz potential of a compact domain in R^m by introducing a renormalization of the r^@a^-^m-potential for @a=<0. This can be considered as generalization of the dual mixed volumes of convex bodies as introduced by Lutwak. We then study the points where the extreme values of the (renormalized) potentials are attained. These points can be considered as a generalization of the center of mass. We also show that only balls give extreme values among bodied with the same volume.

The configuration space of equilateral and equiangular hexagons

We study the configuration space of equilateral and equiangular spatial hexagons for any bond angle by giving explicit expressions of all the possible shapes. We show that the chair configuration is isolated, whereas the boat configuration allows one-dimensional deformations which form a circle in the configuration space.

Ideal, Best Packing, and Energy Minimizing Double Helices

We study optimal double helices with straight axes (or the fattest tubes around them) computationally using three kinds of functionals; ideal ones using ropelength, best volume packing ones, and energy minimizers using two one-parameter families of interaction energies between two strands of types

Energy of knots and the infinitesimal cross ratio

r^{-\alpha}

Regularized Riesz energies of submanifolds

and

Möbius invariant energy of tori of revolution

\frac1r\exp(-kr)

Characterization of balls by generalized Riesz energy

. We compare the numerical results with experimental data of DNA.