The virtual and universal braids

Abstract

We study the structure of the virtual braid group. It is shown that the virtual braid group is a semi-direct product of the virtual pure braid group and the symmetric group. Also, it is shown that the virtual pure braid group is a semi-direct product of free groups. From these results we obtain a normal form of words in the virtual braid group. We introduce the concept of a universal braid group. This group contains the classical braid group and has as its quotient groups the singular braid group, virtual braid group, welded braid group, and classical braid group. Recently some generalizations of classical knots and links were de- fined and studied: singular links (20, 5), virtual links (15, 12) and welded links (10). One of the ways to study classical links is to study the braid group. Singular braids (1, 5), virtual braids (15, 21), welded braids (10) were defined similar to the classical braid group. A theorem of A. A. Markov (4, Ch. 2.2) reduces the problem of classification of links to some alge- braic problems of the theory of braid groups. These problems include the word problem and the conjugacy problem. There are generaliza- tions of Markov's theorem for singular links (11), virtual links, and welded links (14). There are some dierent ways to solve the word problem for the singular braid monoid and singular braid group (8, 7, 22). The solution of the word problem for the welded braid group follows from the fact that this group is a subgroup of the automorphism group of the free group (10). A normal form of words in the welded braid group was constructed in (13). In this paper we study the structure of the virtual braid group V Bn. This is similar to the classical braid group Bn and welded braid group