The L-move and Markov theorems for trivalent braids

TL;DR

This paper extends the L-move approach to trivalent braids, establishing Markov-type theorems and an Alexander's theorem analogue for spatial trivalent graphs, advancing the algebraic and topological understanding of these structures.

Contribution

It introduces a one-move Markov-type theorem for trivalent braids and reformulates it algebraically, expanding the theoretical framework for trivalent braid theory.

Findings

Proved a Markov-type theorem for trivalent braids.

Reformulated the theorem in algebraic terms.

Established an Alexander's theorem analogue for spatial trivalent graphs.

Abstract

The L-move for classical braids extends naturally to trivalent braids. We follow the L-move approach to the Markov Theorem, to prove a one-move Markov-type theorem for trivalent braids. We also reformulate this L-Move Markov theorem and prove a more algebraic Markov-type theorem for trivalent braids. Along the way, we provide a proof of the Alexander's theorem analogue for spatial trivalent graphs and trivalent braids.