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 $\Gamma$-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 smooth energy. Moreover, we show that the unique absolute minimizer of inverse discrete thickness is the regular $n$-gon.