Os grupos de tranças emolduradas e suas generalizações

Abstract

Let n ≥ 2 and let Bn denote the Artin braid group, also known as the braid group of the disk. We denote by FBn the framed braid group. In this thesis, we study framed braid groups and their generalizations. Initially, we develop a structural analysis of the group FBn , investigating several algebraic properties. In particular, we determine its center, lower central series, commutator subgroup, as well as certain Coxeter-type quotients and associated congruence subgroups. Next, we extend our study to the context of surfaces, considering the framed braid groups FBn(M) , where M may be an orientable or non-orientable surface, possibly with a finite number of punctures. Subsequently, we introduce and analyze two generalizations of the framed braid group: the framed virtual braid group FVBn and the framed singular braid group FSGn. For both cases, we present descriptions by generators and relations, and investigate structural properties analogous to those of FBn. Finally, we construct an invariant for singular knots, based on the virtual Temperley–Lieb algebra and the Markov trace, thus establishing a connection between the algebraic theory of braids and the theory of singular knots.