I will define a double affine $Q$-dependent braid group. This group is constructed by appending to the braid group a set of operators $Q_i$, before extending it to an affine $Q$-dependent braid group. I show specifically that the elliptic braid group and the double affine Hecke algebra can be obtained as quotient groups. Complementing this I will also present a pictorial representation of the double affine $Q$-dependent braid group based on ribbons living in a toroid. I graphically describe the action of the operators $Q_i$ and show that in this particular representation $Q$ generates a twist in the ribbon. Subsequently I will show that this graphical representation is also valid for all double affine Hecke algebras.

National University of Ireland, Maynooth