Graphical Calculus for the Double Affine Q-Dependent Braid Group

TL;DR

This paper introduces a new double affine Q-dependent braid group, providing a pictorial ribbon-based representation on a toroid, and links it to elliptic braid groups and DAHA, clarifying the role of parameter q.

Contribution

It constructs the double affine Q-dependent braid group, offers a visual ribbon model, and connects it to known algebraic structures like elliptic braid groups and DAHA.

Findings

Pictorial ribbon representation of the double affine Q-dependent braid group.

Demonstrated that elliptic braid group and DAHA are quotient groups of this new structure.

Connected the parameter q to a twist in the ribbon within the pictorial model.

Abstract

We 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. We show specifically that the elliptic braid group and the double affine Hecke algebra (DAHA) can be obtained as quotient groups. Complementing this we present a pictorial representation of the double affine $Q$-dependent braid group based on ribbons living in a toroid. We show that in this pictorial representation we can fully describe any DAHA. Specifically, we graphically describe the parameter $q$ upon which this algebra is dependent and show that in this particular representation $q$ corresponds to a twist in the ribbon.