Towards the quantum exceptional series

Abstract

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 (in the adjoint representation). If a slight strengthening of Deligne's conjecture on the existence of a (classical) exceptional series is true, then this relation holds for a new two-variable quantum exceptional polynomial, at least as a power series near $q=1$. The single quantum exceptional relation can be viewed as a deformation of the Jacobi relation, and implies a deformation of the Vogel relation that motivated the conjecture on the classical exceptional series. We find a conjectural basis for the space of diagrams with $n$ loose ends modulo the quantum exceptional relation for $n \le 6$, with dimensions agreeing with the classical computations, and compute the matrix of inner products, and the quantum dimensions of idempotents. We use the skein relation to compute the conjectural quantum exceptional polynomial for many knots. In particular we determine (unconditionally) the values of the quantum polynomials for the exceptional Lie algebras on these knots. We can perform these computations for all links of Conway width less than $6$, which includes all prime knots with 12 or fewer crossings. Finally, we prove several specialization results relating our conjectural family to certain quantum group categories, and conjecture a number of exceptional analogues of level-rank duality.