2015/07/09 by Fedoseev, Denis, Manturov, Vassily, Cheng, Zhiyun
#FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.1507.02700
In the present paper, we introduce ℤ2-braids and, more generally, G-braids for an arbitrary group G. They form a natural group-theoretic counterpart of G-knots, see \citereidmoves. 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 \citeparity) 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.