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

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