The $L$-move and Markov theorems for trivalent braids

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 theorem analogue for spatial trivalent graphs and trivalent braids.