  We study a two-point self-avoidance energy $E_q$ which is defined for all
rectifiable curves in $R^n$ as the double integral along the curve of $1/r^q$.
Here $r$ stands for the radius of the (smallest) circle that is tangent to the
curve at one point and passes through another point on the curve, with obvious
natural modifications of this definition in the exceptional, non-generic cases.
It turns out that finiteness of $E_q(\gamma)$ for $q\ge 2$ guarantees that
$\gamma$ has no self-intersections or triple junctions and therefore must be
homeomorphic to the unit circle or to a closed interval. For $q>2$ the energy
$E_q$ evaluated on curves in $R^3$ turns out to be a knot energy separating
different knot types by infinite energy barriers and bounding the number of
knot types below a given energy value. We also establish an explicit upper
bound on the Hausdorff-distance of two curves in $R^3$ with finite $E_q$-energy
that guarantees that these curves are ambient isotopic. This bound depends only
on $q$ and the energy values of the curves. Moreover, for all $q$ that are
larger than the critical exponent $2$, the arclength parametrization of
$\gamma$ is of class $C^{1,1-2/q}$, with H\"{o}lder norm of the unit tangent
depending only on $q$, the length of $\gamma$, and the local energy. The
exponent $1-2/q$ is optimal.
