On some knot energies involving Menger curvature (Q387182)

Let \(\mathcal{C}\) be the class of all closed rectifiable curves \(\gamma \subset \mathbb{R}^3\) whose one-dimensional Hausdorff measure \(\mathcal{H}^1(\gamma)\) is equal to 1. A functional \(\mathcal{E}:\mathcal{C}\to [-\infty,\infty]\) that is finite on all simple smooth loops \(\gamma\in \mathcal{C}\), with the property that \(\mathcal{E}(\gamma_i)\) tends to \(+\infty\) as \(i\to \infty\) on any sequence of simple loops \(\gamma_i \in \mathcal{C}\) that converge uniformly to a limit curve with at least one self-intersection is a self-repulsive or charge energy functional. If \(\mathcal{E}\) is self-repulsive and bounded from below it is called a knot energy. An example of knot energy is the Möbus energy: NEWLINE\[NEWLINE\mathcal{E}_{Mob}= \int^1_0 \int^1_0 \biggl\{ \frac{1}{ |\gamma(s)-\gamma(t)|^2}- \frac {1}{d_\gamma(s,t)^2}\biggr\}ds\,\, dt \,\, \text{for}\,\, \gamma \in \mathcal{C}.NEWLINE\]NEWLINE Here \(d_\gamma (s,t)= \min \{ |s-t|,1-|s-t|\}\) is the intrinsic distance of the two curve points \(\gamma(s), \gamma(t)\). A knot energy \(\mathcal{E}\) is minimizable if in each knot class there is at least one representative in \(\mathcal{C}\) minimizing \(\mathcal{E}\) within this knot class. \(\mathcal{E}\) is called tight if \(\mathcal{E}(\gamma_i)\) tends to \(+\infty\) on a sequence \(\{\gamma_i\}\subset \mathcal{C}\) with a pull-tight phenomenon.NEWLINENEWLINEIn the present paper the authors investigate knot-energetic properties of geometrically defined curvature energies involving Menger curvature. For three distinct points \(x,y,z\in \mathbb{R}^3\), the Menger curvature of these points is the inverse of the circumcircle radius \(R(x,y,z)\). If the points are colinear the Menger curvature is zero. If \(x\neq y\), then \(R(x,y,y)=|x-y|/2\) and \(R(x,x,x)=0\). If the points \(x,y,z\) are on the curve \(\gamma\) one obtains the following invariants NEWLINE\[NEWLINE\rho[\gamma](x,y) =\inf_{\substack{ z\in \gamma; z\neq\\ x\neq y\neq z}} R(x,y,z), \rho_G[\gamma](x)=\inf_{\substack{ y,z\in\gamma\\ z\neq x\neq y\neq z}} R(x,y,z), \Delta[\gamma]=\inf_{\substack{ x,y,z\in\gamma;\\ z\neq x\neq y\neq z}} R(x,y,z).NEWLINE\]NEWLINE Repeated integrations over inverse powers of these radii with respect of remaining variables lead to the various Menger curvature energies: \(\mathcal{M}_p(\gamma),\;\mathcal{I}_p(\gamma),\;\mathcal{U}_p(\gamma)\). The authors study the properties of these energies as well some other energies. They discuss lower semicontinuity of certain of energies, isotopy between several curves \(\gamma\), basic detecting unknots, then isotopies to polygonal lines and crossing number bounds.