Temperley–Lieb link-state standard modules by defect number

Let $R$ be a commutative ring with unit, let $\delta \in R$, and let $n\ge 0$. Let $t$ be an integer with $0\le t\le n$ and $n-t$ even. The diagram module $M_R(n,t)$ has $n$ bottom boundary points and $t$ top boundary points, with stacking and loop removal as in Temperley–Lieb planar diagram data and endomorphism modules. A component of a basis diagram is propagating if it meets both the bottom and top boundaries. Write $\mathrm{prop}(D)$ for the number of propagating components of a diagram $D:n\to t$.

Let $M_R^{<t}(n,t)$ be the $R$-span of the basis diagrams $D:n\to t$ with $\mathrm{prop}(D)<t$. Define the Temperley–Lieb link-state standard module

$W_{n,t}(R,\delta ):=M_R(n,t)\mathrm{/}{M}_R^{<t}(n,t)\mathrm{.}$

Write $T_{n,t}$ for the set of diagrams in $\mathcal{D}(n,t)$ with exactly $t$ propagating components. The cosets of these diagrams form an $R$-basis of $W_{n,t}$. Identifying each diagram with its bottom arcs and defects gives a partial matching of the $n$ ordered bottom sites, with the additional condition that the $t$ unpaired sites are joined to the ordered top sites by noncrossing propagating strands. In particular, no defect lies inside an arc. The unpaired sites are called defects. Thus $t$ is the defect or through-line number; in sources using $p$ for the number of arcs, $p=(n-t)\mathrm{/}2$.

The intended right action of ${\mathrm{T}\mathrm{L}}_n(R,\delta )$ is $x\cdot a=[x\circ a]$, where $x\in M_R(n,t)$, $a\in M_R(n,n)$, and $x\circ a$ stacks $x$ above $a$. The submodule $M_R^{<t}(n,t)$ is stable under this action, so it descends to $W_{n,t}$; the complete well-definedness argument is The link-state quotient is a right Temperley–Lieb module ↗.

The TikZ half-diagram below is the basis link state in $W_{4,2}$ with a bottom arc joining sites $2$ and $3$ and defects at sites $1$ and $4$. The top endpoints label the two propagating strands in their left-to-right order. A picture records this matching; the module and action are defined by the labeled matching and the stacking operation above.