The Framed Braid Group and 3-Manifolds
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Abstract
Abstract. The framed braid group on n strands is defined to be a semidirectproduct of the braid group B„ and Z . Framed braids represent 3-manifoldsin a manner analogous to the representation of links by braids. Consider twoframed braids equivalent if they represent homeomorphic 3-manifolds. Themain result of this paper is a Markov type theorem giving moves that generatethis equivalence relation. In this paper the group of framed braids $n is introduced. This group issimilar to the braid group and is an initial attempt to understand 3-manifoldsin a manner analogous to the braid approach to links in the 3-sphere. The maintheorem describes the equivalence relations on \J^LX 5« that yields the set of 3-manifolds. Framed braids. Let Bn denote the braid group with generators ox, oj, ... , 1; (2) OiOi+xOi = Oi+xOiOi+x. The geometric braid ai is shown in Figure 1.Let X„ denote the symmetric group acting on {1, 2, ... , zz} . Let n : B„ -+Z„ be the quotient map sending a, to the transposition (j, z'+l). The kernel of?i is the pure braid group denoted by P„ . Bn acts on {1, 2, ... , zz} throughn, i.e., aii) = zt((r)(z) for a £ Bn. This paper follows the convention that thesymmetric group acts from the right so that (ot)(z) = t(<7(z')) for a, t £ B„ .Definition. The framed braid group #„ is the group generated by ax, a2, ... ,<7„_i , tx, I2, ... , tn with the relations (1), (2) and additional relations