The Temperley–Lieb link-state diagram basis is cellular

Every diagram $D:n\to n$ has a definite number $t$ of propagating strands, with $0\le t\le n$ and $n-t$ even. Cut each propagating strand at a horizontal midpoint. The upper half, reflected, is a link state $x:n\to t$; the lower half is a link state $y:n\to t$. Conversely, $x^{\ast }\circ y$ joins the $t$ corresponding defect endpoints in order and yields a diagram with exactly $t$ propagating strands. The cuts are determined by $D$, so these constructions are inverse. Thus the diagrams $C_{x,y}^t$ are in bijection with the basis diagrams of ${\mathrm{T}\mathrm{L}}_n$ and hence form an $R$-basis.

Define $\ast$ on every basis diagram by horizontal reflection and extend $R$-linearly. Reflecting a stack reverses its two factors and preserves each closed component, so $(ab{)}^{\ast }=b^{\ast }{a}^{\ast }$; reflecting twice restores the original pairing and $1_n^{\ast }=1_n$. Also $(C_{x,y}^t{)}^{\ast }=C_{y,x}^t$. Thus $\ast$ is the involutive anti-automorphism required by [F3].

For $t\in {\mathrm{\Lambda }}_n$, let $J_t$ be the span of diagrams with fewer than $t$ propagating strands. The stated order makes these exactly the basis elements indexed by labels strictly greater than ${\lambda }_t$, so $J_t=A^{⊳{\lambda }_t}$. Every propagating path in a stacked product must pass through a distinct propagating path in each factor, so a product has no more propagating strands than either factor; therefore $J_t$ is a two-sided ideal. For $a\in {\mathrm{T}\mathrm{L}}_n$, write the quotient of $y\circ a$ in $M_R(n,t)\mathrm{/}{M}_R^{<t}(n,t)$ as ${\sum }_v{r}_{v}v$ in its link-state basis. Then $C_{x,y}^{t}a=x^{\ast }\circ (y\circ a)\equiv {\sum }_v{r}_v{C}_{x,v}^t$ modulo $J_t$. The coefficients $r_v$ are those of $y\cdot a$ and are independent of $x$, so this is exactly the fixed-first-index cellular multiplication rule in [F3].

For link states $y,u$, the middle composite $y\circ u^{\ast }$ is a diagram $t\to t$. If it has all $t$ propagating strands, planarity makes it ${\delta }^c{1}_t$, and the full product is $C_{x,y}^t{C}_{u,v}^t=x^{\ast }\circ (y\circ u^{\ast })\circ v={\varphi }_{n,t}(u,y)C_{x,v}^t={\varphi }_{n,t}(y,u)C_{x,v}^t$, using symmetry of the form. If $y\circ u^{\ast }$ has fewer than $t$ propagating strands, the full product also has fewer than $t$ and lies in $J_t$, so the same congruence holds with coefficient zero. Hence ${\varphi }_{n,t}$ is the cellular cell form. Finally, the generic right cell-module action read from $C_{x,x}^{t}a$ has the same coefficients $r_v$ as $y\cdot a$ in step 2.2, so its basis and action identify with $W_{n,t}$.