An elementary proof that classical braids embed into virtual braids

Abstract

The aim of the present note is to show that the natural map from classical braids to virtual braids is an inclusion; this proof does not use any complete invariants of classical braids; it is based on the projection from virutal braids to classical braids (similar to the one given in \cite{Projection}); this projection is the idenitiy map on the set of virtual braids. The projection is well defined not only for the virtual braid group but also for the quotient group of the virtual (pure) braid group by the so-called virtualization move. The idea of this projection is closely related to the notion of parity and to the groups $G_{n}^{k}$, introduced by the author.