Twisted knot theory, introduced by M. O. Bourgoin, is a generalization of virtual
knot theory. It naturally yields the notion of a twisted braid, which is closely related
to the notion of a virtual braid due to Kauffman. We first prove that any twisted link
can be described as the closure of a twisted braid, which is unique up to
certain basic moves. This is the analogue of the Alexander theorem and the
Markov theorem for classical braids and links. Then we also give reduced
presentations for the twisted braid group and the flat twisted braid group. These
reduced presentations are based on the fact that these twisted braid groups on
strands are
generated by a single braiding element and a single bar element plus the generators of the symmetric
group on
letters.
Keywords
twisted link, twisted braids, braiding link diagram,
reduced presentation
