1. Define a functional, "energy of knots", on the space of the knots.
2. Define the canonical shape of any given knot type
as the embedding that attains the minimum value of that functional
within its ambient isotopy class.
(We call it an energy minimizer.)
For this purpose we consider a functional on the space of knots
that blows up if a knot degenerates to a 'singular knot' with
self-intersections.
Such a functional is called a knot energy functional.
The electrostatic energy of a charged knot would be
the most naive candidate for that.
Although, it blows up for any knot.
I could get rid of this difficulty of explosion by
so called regularization to have a finite valued functional.
But then, this functional does not blow up for
a 'singular knot' with double points.
By changing the power index of the integrand,
which corresponds to a non-physical assumption that
Coulomb's repelling force between a pair of point charges
is inversely proportional to more than the cube of the distance
between them, I defined a family of energy functionals.
My research concerns the problem to determine whether
there exists an energy minimizer
for each knot type with respect to a given knot energy functional.
The answer to this problem depends on the following conditions:
the power index of the integrand of the energy functional,
the metric of the ambient space, i.e. whether
it is Euclidean, spherical, or hyperbolic, and
the knot type, i.e. whether it is prime or not.
Since the end of 1998, I started the joint work with
Prof. Remi Langevin (Universite de Bourgogne, France).
Let us consider 5-dimensional Minkowski space with the Lorentz metric.
The 3-sphere can be considered as the set of points at infinity
in the light cone.
We study properties of knots which are preserved under
the action of the conformal group.
We gave a new interpretation of the Moebius invariance of
the first example of an energy of knots, E, which had been
proved by Freedman, He, and Wang.
E can be expressed in terms of the cross-ratio of four points
x, x+dx, y, and y+dy considered as complex numbers on the Riemann sphere
which is twice tangent to the knot at points x and y.
This cross-ratio, which we called the 'infinitesimal cross-ratio',
is a complex valued 2-form on S^1 times S^1 minus the diagonal set.
The real part of the infinitesimal cross-ratio is (up to a constant) equal to
the pull-back of the standard symplectic form of the cotangent bundle
of the 2-sphere under the idendification of S^2 times S^2 minus
the diagonal set and the total space of the cotangent bundle of S^2.
This interpretation gives us another interpretation of the energy E.
These results, together with related topics, can be found in my book [11].