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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:\ud \ud 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.\ud \ud 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 .\ud \ud 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.
• If now c is of type (i) ,i.e. c E 'Hn-.(q, Q) and b is of type (ii) , i.e. b = X9n-lY , x, Y E 'Hn-1(q, Q), then: t r (9nX9n-lY9nC) B=.r t r (X9n9n-19nYC ) B=.r t r (X9n-19n9n-lYC ) (=3') [1] [2] [4] [5] [6] [7] [8] [9] [13] [14] [15] J.W. Alexander, A lemma on systems of knotted curves, Proe. Nat. Aead. SeL U.S.A. 9 , pp. 93-95, 1923.
Soc. 30 , pp. 275-306, 1928.
[3] J.W. Alexander, G.B. Briggs On types of knotted curves, Annals of Mathematics 2 , vol. 28, pp. 563-586, 1927.
E. Artin, Theorie der Zopfe, Abh. Math. Sem. Univ. Hamburg 4 , pp. 47-72, 1926.
E. Artin, Theory of braids, Ann. of Math. 48 , pp. 101-126,643-649, 1947.
D. Bennequin, Entrlacements et equations de Pfaffe, Asterisque 107-108 , pp.
82 , Princeton University Press, Princeton 1974.
N. Bourbaki, Groupes et algebres de Lie, Chap. 4-6 Hermann, Paris, 1968.