Menger curvature as a knot energy

Motivated by the suggestions of Gonzalez and Maddocks, and Banavar et al. to use geometrically defined curvature energies to model self-avoidance phenomena for strands and sheets we give a self-contained account, aimed at non-experts, on the state of art of the mathematics behind these energies. The basic building block, serving as a multipoint potential, is the circumradius of three points on a curve. The energies we study are defined as averages of negative powers of that radius over all possible triples of points along the curve (or via a mixture of averaging and maximization). For a suitable range of exponents, above the scale invariant case, we establish self-avoidance and regularizing effects and discuss various applications in geometric knot theory, as well as generalizations to surfaces and higher-dimensional submanifolds.