Oleg Karpenkov
Energies of knots and graphs.

Energies of knots are functionals on the space of smooth or
piecewise smooth knots, it depends on the type of energy. We
consider the best-known energies of knots: Mobius energy, energy
of osculating circles and some other energy functionals, and also
we present a new Mm-energy of a knot. The most important
properties of the energies are continuity, positivity, and the
property that the energy tends to infinity while the knot tends
to some knot with self-crossings. This properties of energies
give another approach to the problem of identification the knot
type for some given knot based on variational principles. The
described approach can be generalized to the case of graphs for
some special types of energies. We will study Mm-energy of graphs
more detailed.