Some hecke-type algebras derived from the braid group with two fixed strands

Abstract

We construct some Hecke-type algebras, and most notably the quotient algebra H2,n(q) of the group-algebra Z[q±1]B2,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 groups B2,n are known to be related to the knot theory of certain families of 3-manifolds, and the algebras H2,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 H2,n(q), and we have singled out a subset Λn of this algebra for which we proved it is a spanning set, whereas ongoing research aims at proving it to be a basis.