Jones polynomial of tangles using Temperley-Lieb algbra? 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.
 A: Yes, the same thing can be done using in terms of the TL algebra. Namely, that is how the Jones polynomial was originally defined. For TL, there are two ways to get the link invariant, which both leads to a skein theory formula. The original way of Jones or the Kaufmann bracket. See also here: https://math.berkeley.edu/~vfr/jones.pdf
Another way to see it is, that the representation category of $\mathrm{SU(2)}_k$ (which is the same of $U_q\mathrm{sl}_2$ for some $q$) and the Temperley-Lieb-Jones planar algebra $A_{k+1}$ are essentially the same, just that in
the $\mathrm{SU(2)}_k$ case the fundamental representation is pseudo-real while Temperley-Lieb is real, but they lead to the same link invariant (at different discrete values I think).
The issue is also considered in for example, where they show how the Skein relation construction and the quantum group gives the same invariant up to a minus sign, or exactly the same using a non-standard ribbon element and for framed undirected links: 
http://arxiv.org/abs/1002.0555
