Cactus Doodles

TL;DR

This paper introduces cactus doodles, a new class of combinatorial and geometric objects related to cactus groups, defined via local moves on plane curves, and explores their fundamental properties and connections to algebraic structures.

Contribution

It defines cactus doodles through local moves, links them to cactus groups via a closing procedure, and investigates their basic properties, expanding the understanding of these mathematical objects.

Findings

Cactus doodles can be obtained from cactus group elements.

They can be described using local moves on plane curves.

Basic properties of cactus doodles are established.

Abstract

Cactus doodles are combinatorial/geometric objects that are related to cactus groups in the same way as knots are related to braids. We define them in terms of local moves on plane curves, show that they can be obtained from elements of the cactus group by a "closing" procedure and establish some of their basic properties.