Variational formulae and estimates of O’Hara’s knot energies

Abstract

O’Hara’s energies, introduced by Jun O’Hara, were proposed to answer the question what the canonical shape in a given knot type is, and were configured so that the less the energy value of a knot is, the “better” its shape is. The existence and regularity of minimizers has been well studied. In this paper, we calculate the first and second variational formulae of the [Formula: see text]-O’Hara energies and show absolute integrability, uniform boundedness, and continuity properties. Although several authors have already considered the variational formulae of the [Formula: see text]-O’Hara energies, their techniques do not seem to be applicable to the case [Formula: see text]. We obtain the variational formulae in a novel manner by extracting a certain function from the energy density. All of the [Formula: see text]-energies are made from this function, and by analyzing it, we obtain not only the variational formulae but also the estimates in several function spaces.