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First off I think there's at least one other knot polynomial satisfying these skein relations: the Reshetikhin-Turaev invariant coming from the 133-dimensional representation of E7. Unfortunately for you, I don't think anyone's ever given an elementary description of that knot polynomial. On the other hand, it would be nice (and, in my opinion, publishable) to see a purely elementary description of this RT invariant, so if you find a knot polynomial which you can prove exists by elemenatary means you should feel free to contact me and I'll let you know if it's E7.

I may have missed something (I only did a quick heuristic search), but I suspect that this E7 example is the only other RT invariant which satisfies this sort of skein relation (other than the ones that you already listed). [Update: there's also the spin representation of Spin(12).] I expect that there are no known knot polynomials satisfying this skein relation which don't come from RT... I'd have to do some more checking to be totally sure...

On the other hand, if I understand everything correctly, according to the introduction of this fascinating (though somewhat mysterious) paper your cubic skein relation is not enough to define its value on all links. They claim that this result is proved here, but I'm a little confused as Dabkowski and Przytycki result seems to me to be slightly weaker. That is, it seems to me that they're only proving that the most natural way you might prove that you can evaluate all links using this relation doesn't work. However, I might be missing something here. At any rate, I would not be very optimistic about that skein relation being enough to reduce everything.

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First off I think there's at least one other knot polynomial satisfying these skein relations: the Reshetikhin-Turaev invariant coming from the 133-dimensional representation of E7. Unfortunately for you, I don't think anyone's ever given an elementary description of that knot polynomial. On the other hand, it would be nice (and, in my opinion, publishable) to see a purely elementary description of this RT invariant, so if you find a knot polynomial which you can prove exists by elemenatary means you should feel free to contact me and I'll let you know if it's E7.

I may have missed something (I only did a quick heuristic search), but I suspect that this E7 example is the only other RT invariant which satisfies this sort of skein relation (other than the ones that you already listed). I expect that there are no known knot polynomials satisfying this skein relation which don't come from RT... I'd have to do some more checking to be totally sure...

On the other hand, if I understand everything correctly, according to the introduction of this fascinating (though somewhat mysterious) paper your cubic skein relation is not enough to define its value on all links. They claim that this result is proved here, but I'm a little confused as Dabkowski and Przytycki result seems to me to be slightly weaker. That is, it seems to me that they're only proving that the most natural way you might prove that you can evaluate all links using this relation doesn't work. However, I might be missing something here. At any rate, I would not be very optimistic about that skein relation being enough to reduce everything.