Energy functionals and normal forms of knots and plane curves

Abstract

In this paper, a new theory, called theory of thickened knots, is described. Roughly speaking, a thickened knot is a tube smoothly embedded in Euclidean space considered up to isotopy. The theory depends on a parameter ε ∈ ( 0 , 1 ] \varepsilon \in (0,1] , which is equal to the diameter of the two-dimensional disk that sweeps out the tube. It is shown that for each fixed ε \varepsilon , there is only a finite number of nonisotopic knots. An approach to the solution of the classification problem of thickened knots, based on the idea of knot energy, is suggested. Concrete energy functionals for thickened knots are proposed, their local minima yield certain positions of the given knot which are called normal forms. Experiments with wire models of thickened knots are described, and they are in agreement with the theory. A number of promising conjectures are formulated.