Decomposition of generalized O'Hara's energies

Abstract

O'Hara introduced several functionals as knot energies. One of them is the M\"{o}bius energy. We know its M\"{o}bius invariance from Doyle-Schramm's cosine formula. It is also known that the M\"{o}bius energy was decomposed into three components keeping the M\"{o}bius 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. On the way to derive it, we obtain an analogue of the cosine formula for the generalized O'Hara energy. Furthermore, using decomposition, the first and second variational formulae are derived.