Synchronization of Kuramoto Oscillators on Knots

TL;DR

This paper explores the synchronization behavior of oscillators placed at crossings of knot diagrams using the Kuramoto model, introducing order parameters to analyze different regions of the knot.

Contribution

It proposes a novel approach to study oscillator synchronization on knots by defining order parameters based on knot diagrams and their regions.

Findings

Order parameters effectively characterize synchronization states.

Different parts of the knot influence synchronization patterns.

Method provides new insights into topological effects on oscillator dynamics.

Abstract

A knot is a circle embedded in the space. Projecting a knot on a plane, we obtain a diagram which is known as the knot diagram. The vertices of the diagram, where the curved lines are crossed, can be considered as sites occupied by oscillators. The synchronization of these oscillators can be studied by means of a Kuramoto model. Here we propose to define some order parameters, of the complete knot diagram and of its regions, to study the synchronization of the system with regard to the different parts of it.