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O'Hara's energies, introduced by Jun O'Hara, were proposed to answer thequestion of what is the canonical shape in a given knot type, and wereconfigured so that the less the energy value of a knot is, the "better" itsshape is. The existence and regularity of minimizers has been well studied. Inthis article, we calculate the first and second variational formulae of the$(\alpha,p)$-O'Hara energies and show absolute integrability, uniformboundedness, and continuity properties. Although several authors have alreadyconsidered the variational formulae of the $(\alpha,1)$-O'Hara energies, theirtechniques do not seem to be applicable to the case $p>1$. We obtain thevariational formulae in a novel manner by extracting a certain function fromthe energy density. All of the $(\alpha,p)$-energies are made from thisfunction, and by analyzing it, we obtain not only the variational formulae butalso estimates in several function spaces.

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