The definition of the Jones polynomial of tangles (a la Reshetikhin and Turaev) uses the space of invariants for $U_q sl_2$ and R-matrices. It seems to me the same thing cane be done in terms of the Temperley-Lieb algebra as well (because the Jones polynomial of links can be defined in its terms) but I can't find a reference.

This can be done, for example if we can assign a homomorphism $TL_n\to TL_{n-2}$ to a cap tangle. (As a guess we can take a flat $(n,n)$-tangle $T$ (which is a bais element for $TL_n$) and then erase the connected components of $T$ which would be connected to each other by our given cap, to get an element of $TL_{n-2}$.)

The reason I ask this question is that using Temperley-Lieb algebra may be conceptually simpler.