A spanning set and potential basis of the mixed Hecke algebra on two fixed strands

The mixed braid groups $B_{2,n}, \ n \in \mathbb{N}$, with two fixed strands and $n$ moving ones, are known to be related to the knot theory of certain families of $3$-manifolds. In this paper we define the mixed Hecke algebra $\mathrm{H}_{2,n}(q)$ as the quotient of the group algebra ${\mathbb Z}\, [q^{\pm 1}] \, B_{2,n}$ over the quadratic relations of the classical Iwahori-Hecke algebra for the braiding generators. We furhter provide a potential basis $\Lambda_n$ for $\mathrm{H}_{2,n}(q)$, which we prove is a spanning set for the $\mathbb{Z}[q^{\pm 1}]$-additive structure of this algebra. The sets $\Lambda_n,\ n \in \mathbb{Z}$ appear to be good candidates for an inductive basis suitable for the construction of Homflypt-type invariants for knots and links in the above $3$-manifolds.