A note on the pure cactus group of degree three and the configuration space of four points on the circle

TL;DR

This paper explores the relationship between the pure cactus group of degree three and the configuration space of four points on a circle, providing explicit descriptions to deepen understanding of their connection.

Contribution

It offers an explicit description of the relationship between the pure cactus group of degree three and the configuration space of four points on the circle, a novel clarification in the field.

Findings

Explicit description of the pure cactus group of degree three

Clarification of its relationship with the configuration space of four points

Enhanced understanding of cactus groups in geometric contexts

Abstract

The cactus group was introduced by Henriques and Kamnitzer as an analogue of the braid group. In this note, we provide an explicit description of the relationship between the pure cactus group of degree three and the configuration space of four points on the circle.