Some Hecke-Type Algebras Derived from the Braid Group with Two Fixed Strands

We construct some Hecke-type algebras, and most notably the quotient algebra H2, n(q) of the group-algebra Z[q (+/- 1)] B-2,B-n of the mixed braid group B2, n with two identity strands and n moving ones, over the quadratic relations of the classical Hecke algebra for the braiding generators. The groupsB(2,n) are known to be related to the knot theory of certain families of 3-manifolds, and the algebras H-2, n(q) are aimed for the construction of invariants of oriented knots and links in these manifolds. To this end, one needs a suitable basis of H-2, n(q), and we have singled out a subset.n of this algebra for which we proved it is a spanning set, whereas ongoing research aims at proving it to be a basis.