Tangent-point self-avoidance energies for curves

TL;DR

This paper introduces a self-avoidance energy for curves that prevents self-intersections, classifies knot types by energy barriers, and provides bounds on curve similarity based on energy, with implications for knot theory and geometric analysis.

Contribution

It defines a new tangent-point self-avoidance energy for curves, analyzes its properties, and establishes bounds and regularity results related to knot classification and curve isotopy.

Findings

Finite energy implies no self-intersections or triple junctions.

For q>2, the energy separates different knot types by infinite barriers.

Explicit bounds on Hausdorff distance ensure ambient isotopy of curves.

Abstract

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.