A discretization of O'Hara's knot energy and its convergence

TL;DR

This paper introduces a discrete version of O'Hara's knot energy for polygons, proving its convergence to the continuous energy and establishing its $ ext{Gamma}$-convergence, which is crucial for understanding minimizers.

Contribution

It develops a discrete formulation of O'Hara's knot energy and proves its convergence and $ ext{Gamma}$-convergence to the continuous energy, advancing the mathematical understanding of knot energies.

Findings

Discrete energy values converge to the continuous O'Hara's energy.

Discrete energy converges in the sense of $ ext{Gamma}$-convergence.

Analysis of minimality of the discrete energy.

Abstract

In this paper, we propose a discrete version of O'Hara's knot energy defined on polygons embedded in the Euclid space. It is shown that values of the discrete energy of polygons inscribing the curve which has bounded O'Hara's energy converge to the value of O'Hara's energy of its curve. Also, it is proved that the discrete energy converges to O'Hara's energy in the sense of $\Gamma$-convergence. Since $\Gamma$-convergence relates to minimizers of a functional and discrete functionals, we need to investigate the minimality of the discrete energy.