Alexander- and Markov-type theorems for virtual trivalent braids

TL;DR

This paper establishes Alexander- and Markov-type theorems for virtual spatial trivalent graphs and braids, expanding the theoretical framework of knot theory to more complex virtual structures.

Contribution

It introduces two versions of the Markov-type theorem for virtual trivalent braids, one algebraic and one based on L-moves, advancing the understanding of virtual knot theory.

Findings

Proved Alexander-type theorems for virtual spatial trivalent graphs.

Developed two Markov-type theorems, algebraic and L-move based.

Extended classical braid theory to virtual trivalent structures.

Abstract

We prove Alexander- and Markov-type theorems for virtual spatial trivalent graphs and virtual trivalent braids. We provide two versions for the Markov-type theorem: one uses an algebraic approach similar to the case of classical braids and the other one is based on L-moves.