Does finite knot energy lead to differentiability? Preprint no. 12, Institut für Mathematik, RWTH Aachen 2006; see http://www.instmath.rwth-aachen.de/→ preprints. To appear in J. Knot Theory Ramifications. [4

In this article, we raise the question if curves of finite (j, p)-knot energy introduced by O’Hara are at least pointwise differentiable. If we exclude the highly singular range (j − 2)p ≥ 1, the answer is no for jp ≤ 2 and yes for jp> 2. In the first case, which also contains the most prominent example of the Möbius energy (j = 2, p = 1) investigated by Freedman, He and Wang, we construct counterexamples. For jp> 2, we prove that finite-energy curves have in fact a Hölder continuous tangent with Hölder exponent 1 2 (jp−2)/(p+2). Thus, we obtain a complete picture as to what extent the (j, p)-energy has self-avoidance and regularizing effects for (j, p) ∈ (0,∞)×(0,∞). We provide results for both closed and open curves