On Marked Braid Groups

TL;DR

This paper introduces $ ext{Z}_2$-braids and $G$-braids, extending classical braid groups with additional decorations, generalizing parity concepts, and exploring their algebraic properties and connections to $G$-knots.

Contribution

It defines new classes of decorated braid groups, generalizes parity to $G$-braids, and introduces dotted braid groups as natural extensions of classical braid groups.

Findings

Defined $ ext{Z}_2$-braids and $G$-braids as group-theoretic counterparts of $G$-knots.

Established the concept of dotted braid groups and their relation to classical braid groups.

Presented open problems related to the structure and properties of these new braid groups.

Abstract

In the present paper, we introduce $\mathbb{Z}_2$-braids and, more generally, $G$-braids for an arbitrary group $G$. They form a natural group-theoretic counterpart of $G$-knots, see \cite{reidmoves}. The underlying idea, used in the construction of these objects --- decoration of crossings with some additional information --- generalizes an important notion of {\it parity} introduced by the second author (see \cite{parity}) to different combinatorically--geometric theories, such as knot theory, braid theory and others. These objects act as natural enhancements of classical (Artin) braid groups. The notion of dotted braid group is introduced: classical (Artin) braid groups live inside dotted braid groups as those elements having presentation with no dots on the strands. The paper is concluded by a list of unsolved problems.