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Regularity theory for tangent-point energies: The non-degenerate sub-critical case

In this article we introduce and investigate a new two-parameter family of knot energies $TP^{(p,q)}$ that contains the tangent-point energies. These energies are obtained by decoupling the exponents in the numerator and denominator of the integrand in the original definition of the tangent-point energies. We will first characterize the curves of finite energy in the sub-critical range $p\in(q+2,2q+1)$ and see that those are all injective and regular curves in the Sobolev-Slobodeckiĭ space $W^{(p-1)/q,q}$. We derive a formula for the first variation that turns out to be a non-degenerate elliptic operator for the special case $q=2$ --- a fact that seems not to be the case for the original tangent-point energies. This observation allows us to prove that stationary points of $TP^{(p,2)}$ + λlength, $p\in(4,5)$, λ> 0, are smooth --- so especially all local minimizers are smooth.