We introduce and begin the study of new knot energies defined on knot
diagrams. Physically, they model the internal energy of thin metallic solid
tori squeezed between two parallel planes. Thus the knots considered can
perform the second and third Reidemeister moves, but not the first one. The
energy functionals considered are the sum of two terms, the uniformization term
(which tends to make the curvature of the knot uniform) and the resistance term
(which, in particular, forbids crossing changes). We define an infinite family
of uniformization functionals, depending on an arbitrary smooth function f
and study the simplest nontrivial case f(x)=x2, obtaining neat normal forms
(corresponding to minima of the functional) by making use of the Gauss
representation of immersed curves, of the phase space of the pendulum, and of
elliptic functions