On Marked Braid Groups

Denis Fedoseev; Vassily Manturov; Zhiyun Cheng

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arXiv · 1507.02700

On Marked Braid Groups

Denis Fedoseev·Vassily Manturov·Zhiyun Cheng

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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.

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BibTeX RIS

Denis Fedoseev, Vassily Manturov, Zhiyun Cheng. 2015-07-09. On Marked Braid Groups. https://arxiv.org/abs/1507.02700

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