# 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.

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).
• Yes, sure, braids are tangles. But there are tangles which are not braids namely those poor little caps which have $n$ incoming and $n-2$ outgoing endpoints. For those one needs to define a partial trace on the algebra. – Reza Rezazadegan Dec 27 '14 at 11:08