  The hamiltonian of the $N$-state superintegrable chiral Potts (SICP) model is
written in terms of a coupled algebra defined by $N-1$ types of Temperley-Lieb
generators. This generalises a previous result for $N=3$ obtained by J. F.
Fjelstad and T. M\r{a}nsson [J. Phys. A {\bf 45} (2012) 155208]. A pictorial
representation of a related coupled algebra is given for the $N=3$ case which
involves a generalisation of the pictorial presentation of the Temperley-Lieb
algebra to include a pole around which loops can become entangled. For the two
known representations of this algebra, the $N=3$ SICP chain and the staggered
spin-1/2 XX chain, closed (contractible) loops have weight $\sqrt{3}$ and
weight $2$, respectively. For both representations closed (non-contractible)
loops around the pole have weight zero. The pictorial representation provides a
graphical interpretation of the algebraic relations. A key ingredient in the
resolution of diagrams is a crossing relation for loops encircling a pole which
involves the parameter $\rho= e^{ 2\pi \mathrm{i}/3}$ for the SICP chain and
$\rho=1$ for the staggered XX chain. These $\rho$ values are derived assuming
the Kauffman bracket skein relation.
