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We find a single two-parameter skein relation on trivalent graphs, the quantum exceptional relation, that specializes to a skein relation
associated to each exceptional Lie algebra. Based on Deligne's
conjecture for the (classical) exceptional conjecture, we conjecture
that this relation determines a new two-variable quantum execptional
polynomial. We can compute this two-variable polynomial for all knots
with up to 12 crossings, in particular determining (unconditionally)
the 1-variable polynomial associated to these knots for any of the
exceptional Lie algebras.

This is joint work with Scott Morrison and Noah Snyder.