Approximating Ropelength by Energy Functions

Discrete Thickness

We investigate the relationship between a discrete version of thickness and its smooth counterpart. These discrete energies are defined on equilateral polygons with $n$ vertices. It will turn out that the smooth ropelength, which is the scale invariant quotient of length divided by thickness, is the $Γ$-limit of the discrete ropelength for $n\to\infty$, regarding the topology induced by the Sobolev norm $||\cdot||_{W^{1,\infty}(\mathbb{S}_{1},\mathbb{R}^{d})}$. This result directly implies the convergence of almost minimizers of the discrete energies in a fixed knot class to minimizers of the …

Moebius energy of thick knots

The Moebius energy of a knot is an energy functional for smooth curves based on an idea of self-repelling. If a knot has a thick tubular neighborhood, we would intuitively expect the energy to be low. In this paper, we give explicit bounds for energy in terms of the ropelength of the knot, i.e. the ratio of the length of a thickest tube to its radius.

Polygonal approximation and energy of smooth knots

We establish a fundamental connection between smooth and polygonal knot energies, showing that the Minimum Distance Energy for polygons inscribed in a smooth knot converges to the Moebius Energy of the smooth knot as the polygons converge to the smooth knot. However, the polygons must converge in a ``nice'' way, and the energies must be correctly regularized. We determine an explicit error bound between the energies in terms of the number of the edges of the polygon and the Ropelength of the smooth curve.

拓扑链环与锁子甲网络能量最小化的几何考量

Geometric considerations for energy minimization of topological links and chainmail networks

本文以少量参数刻画 Möbius 与 Minimum Distance 能量，求解 Hopf 链环、Borromean 环及锁子甲网络的最小能量几何，并给出绳长下界猜想与网络效率比较。