Twisted braids

Twisted knot theory, introduced by M. O. Bourgoin, is a generalization of virtualknot theory. It naturally yields the notion of a twisted braid, which is closelyrelated to the notion of a virtual braid due to Kauffman. We first prove that anytwisted link can be described as the closure of a twisted braid, which is uniqueup to certain basic moves. This is the analogue of the Alexander theorem andthe Markov theorem for classical braids and links. Then we also give reducedpresentations for the twisted braid group and the flat twisted braid group. Thesereduced presentations are based on the fact that these twisted braid groups onnstrands are generated by a single braiding element and a single bar element plusthe generators of the symmetric group onnletter.