Decomposition of generalized O’Hara’s energies

Abstract

O’Hara introduced several functionals as knot energies. One of them is the Mobius energy. We know its Mobius invariance from Doyle-Schramm’s cosine formula. It is also known that the Mobius energy was decomposed into three components keeping the Mobius invariance. The first component of decomposition represents the extent of bending of the curves or knots, while the second one indicates the extent of twisting. The third one is an absolute constant. In this paper, we show a similar decomposition for generalized O’Hara energies. We also extend the cosine formula for the Mobius energy to generalized O’Hara energies. It gives us a condition for which the right circle minimizes the energy under the length-constraint. Furthermore, it shows us how far the energy is from the Mobius invariant property. Using decomposition, the first and second variation formulae are derived.