# Cubic skein relations

Hi,

The most generic cubic skein relation for a knot polynome would be

where w^3 is one positive writhe unit. The form is fairly obvious from some self-consistency demands. Now since 20 years or so I try to prove that this already IS a knot polynome but I can't even prove that it is defined for all links, let alone that this relation is self-consistent.

Lately, I used Kauffmans abstract tensor approach to classify "all" S matrix solutions (when you have a state model, the proof is in the computing). I found also some S matrices included having a cubic skein relation, some maybe yet unknown.

The above relation would generalize a) Kauffman AND Dubrovnik polynome, b) the product Jones(x)Jones(y), c) the Kuperberg G2 spider and d) of course all solutions I mentioned above.*

Now sordidly I'm a complete amateur, and if yesterday a paper appeared proving my hypothesis, I might not even recognize THAT it does, let alone find it in the literature.

So, this is my question: Do you know of additional knot polynomes with cubic skein relations, possibly falling under d)? Or maybe even a compendium of all known polynomes? One even an amateur can understand? (E.g. Reshitikhin/Turaev definitely goes over my head.)

If I even write a paper on my S matrix work, of course I'd like to at least identify the solutions already known.

Hauke Reddmann

• w=z^5,u-z^14+z^2-z^-6,v=z^6-z^-2+z^-14 is an example. It also pops up in a possible generalization of the B2 spider as I very recently found.
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I can't see the diagram. –  Qiaochu Yuan Dec 20 '10 at 18:20
@OP: I hope you don't mind; I fixed the link to the diagram. –  Daniel Litt Dec 20 '10 at 19:28
Which now seems to be down again. –  Daniel Litt Dec 20 '10 at 19:38
Do you really want 2 equals signs there? The square of a crossing is invertible, it's typically not 0. –  Noah Snyder Dec 20 '10 at 22:33
I still can't see the diagram –  Paolo Aceto Dec 21 '10 at 0:20