2 6 M ay 2 00 4 Knot theory related to generalized and cyclotomic Hecke algebras of type

In [12] is established that knot isotopy in a 3-manifold may be interpreted in terms of Markov braid equivalence and, also, that the braids related to the 3-manifold form algebraic structures. Moreover, the sets of braids related to the solid torus or to the lens spaces L(p, 1) form groups, which are in fact the Artin braid groups of type B. As a consequence, in [12, 13] appeared the first construction of a Jones-type invariant using Hecke algebras of type B, and this had a natural interpretation as an isotopy invariant for oriented knots in a solid torus. In a further ‘horizontal’ development and using a different technique we constructed in [8] all such solid torus knot invariants derived from the Hecke algebras of type B. Furthermore, in [7] all Markov traces related to the Hecke algebras of type D were consequently constructed.