Orbits by the up-down action of braid diagrams

TL;DR

This paper investigates the orbits of tuples of integers under the action of virtual and classical braid diagrams, providing a detailed classification and conditions for certain colorings, advancing understanding of braid diagram dynamics.

Contribution

It explicitly determines the orbits of all tuples under the up-down action of braid diagrams and characterizes isotropy submonoids and coloring conditions.

Findings

Classified all orbits of integer tuples under braid diagram actions

Identified conditions for braid diagrams to admit up-down colorings

Analyzed the isotropy submonoid structure for braid diagrams

Abstract

The set of all virtual or classical braid diagrams forms a monoid and gives a natural monoid action on a direct product of ${\mathbb Z}$ called the up-down action. In this paper, we determine the orbit of every tuple of ${\mathbb Z}$ under the up-down action of virtual or classical braid diagrams. Moreover, we determine the orbit for irreducible braid diagrams. We also consider the isotropy submonoid and give a condition for a braid diagram to admit an up-down coloring to its closure.