A study of braids in 3-manifolds

This work provides the topological background and a preliminary study for the analogue of the 2-variable Jones polynomial as an invariant of oriented links in arbitrary 3- manifolds via normalized traces of appropriate algebras, and it is organized as follows: Chapter 1: Motivated by the study of the Jones polynomial, we produce and present a new algorithm for turning oriented link diagrams in S3 into braids. Using this algorithm we then provide a new, short proof of Markov's theorem and its relative version. Chapter 2: The objective of the first part of Chapter 2 is to state and prove an analogue of Markov's theorem for oriented links in arbitrary 3-manifolds. We do this by modifying first our algorithm, so as to produce an analogue of Alexander's theorem for oriented links in arbitrary 3-manifolds. In the second part we show that the study of links (up to isotopy) in a 3-manifold can be restricted to the study of cosets of the braid groups Bn,m, which are subgroups of the usual braid groups Bn+m . Chapter 3: In this chapter we try to use the above topological set-up in a procedure analogous to the way V.F.R. Jones derived his famous link invariant. The analogy amounts to the following: We observe that Bn,1 - the braid group related to the solid torus and to the lens spaces L(p, 1) - is the Artin group of the Coxeter group of Bn-type. This implies the existence of an epimorphism of eEn,1 onto the Hecke algebra of Bn-type. Then we give an analogue of Ocneanu's trace function for the above algebras. This trace, after being properly normalized, yields a HOMFLY-PTtype isotopy invariant for oriented links inside a solid torus. Finally, by forcing a strong condition, we normalize this trace, so as to obtain a link invariant in SI x S2