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

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

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: http://arxiv.org/abs/1002.0555

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