# Random discretization of O'Hara knot energy

## TL;DR

This paper introduces a novel random discretization method for O'Hara energy, a knot energy used to define standard shapes of knots, and proves convergence properties using optimal transport theory.

## Contribution

It presents a new random discrete approximation of O'Hara energy and establishes local uniform convergence and compactness results, advancing the analysis of knot energies.

## Key findings

- Established local uniform convergence of the discretization
- Proved compactness of the discrete energy in a transport-based space
- Extended understanding of discretization methods for knot energies

## Abstract

We considered random discrete approximation of O'Hara energy. O'Hara energy is the energy defined for a knot, and O'Hara energy was introduced for defining the standard shape for each knot class (equivalence class by ambient isotopy) by variational method. In the case of a specific exponent, due to energy invariance under Moebius transformation, this energy is called Moebius energy. Although discretization for various Moebius energies has been defined to analyse the shape of the minimizer so far, only Gamma-convergence to the original energy has been shown for a conventional discretization. In this study, we are successful to show locally uniform convergence and compactness of discrete energy in a space based on optimal transport theory, by introducing random discrete approximation of O'Hara energy using random variable.