  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.
