# Quantum E6/E7 knot polynomials

Has anybody seen seen quantum knot invariants associated to (E6, 27) or (E7, 56) worked out in the literature? Even for just simple knots like the trefoil or figure-8?

I suspect these haven't been worked out, but if anybody knows of a reference containing these or even just discussing them, please let me know.

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It is certainly possible to evaluate these for links which can be obtained as the closure of a braid with at most three strings. –  Bruce Westbury Jan 22 '12 at 22:17
I'm pretty sure that the quantum groups Mathematica package which is part of the Knot Atlas package (katlas.org/wiki/Main_Page) can do this. It'll probably be pretty slow. You can ask Scott Morrison if you want more info. –  Noah Snyder Jan 22 '12 at 22:17
I spoke to Scott a couple of days ago. His package cannot compute quantum invariants for F4 or En due to the complexity of some intermediate computations. He noted that one could avoid these issues by, e.g. writing down the quantum positive roots in terms of the PBW bases... working this out is [currently] a bit beyond my primitive background in QA. However, if anybody knows of a good source for such things, that would be equally useful. –  Ross Elliot Jan 22 '12 at 22:38
Ok, my bad. I'd used it for some more basic quantum En calculations before, but I guess they were all a lot simpler than what you need. –  Noah Snyder Jan 23 '12 at 0:15
I would write down the representations of the braid group directly. –  Bruce Westbury Jan 23 '12 at 6:59

$(E7, 56)$ belongs to a series for which I computed some skein relations in my phd thesis (see p55-56 of http://web.univ-ubs.fr/lmam/patureau/articles/these.ps.gz with $Y=X^{19}$). Unfortunately the set of skein relations is not complete but you can use it to compute the quantum invariants of small knots.

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