The hamiltonian of the N-state superintegrable chiral Potts chain can be written in terms of a coupled algebra defined by N - 1 types of Temperley-Lieb generators.
I will present a pictorial representation of this coupled algebra for the N=3 case which involves a generalisation of the well known pictorial representation of the Temperley-Lieb algebra to include a line or pole around which loops can become entangled. The pictorial representation provides a graphical proof of the algebraic relations. A crucial ingredient in the resolution of diagrams is a crossing relation for loops encircling a line.
This pictorial representation also applies to a staggered XX spin chain used as a basic model (the Su-Schrieffer-Heeger model) for a topological insulator. It is anticipated that this algebraic/pictorial approach may lead to further progress in understanding various aspects of the superintegrable chiral Potts model.