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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.

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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

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  • $\begingroup$ Thank you, I didn't know about the sign problem. But this still doesn't answer my question. As far as I know the original definition f the Jones polynomial was given in terms of a braid group representation on Hecke algebras so it didn't have much to do with tangles. $\endgroup$ – Reza Rezazadegan Dec 24 '14 at 11:34
  • $\begingroup$ The definition of Jones polynomial from TL algebras uses a trace on the algebra which is given by closing up the element and counting the number of resulting circles. One may define a "partial trace" by closing up only two connected components of this tangle to get the homomorphism I alluded to in my question. I should verify if this gives the right answer. $\endgroup$ – Reza Rezazadegan Dec 24 '14 at 11:41
  • $\begingroup$ I would say it has a lot to do with tangles. An element in the braid group represents a tangle and taking the trace it represents a link, see figure 1 in ams.org/journals/bull/1985-12-01/S0273-0979-1985-15304-2 $\endgroup$ – Marcel Bischoff Dec 24 '14 at 17:11
  • $\begingroup$ 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. $\endgroup$ – Reza Rezazadegan Dec 27 '14 at 11:08

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