  We construct link invariants using the $D_{2n}$ subfactor planar algebras,
and use these to prove new identities relating certain specializations of
colored Jones polynomials to specializations of other quantum knot polynomials.
These identities can also be explained by coincidences between small modular
categories involving the even parts of the $D_{2n}$ planar algebras. We discuss
the origins of these coincidences, explaining the role of $SO$ level-rank
duality, Kirby-Melvin symmetry, and properties of small Dynkin diagrams. One of
these coincidences involves $G_2$ and does not appear to be related to
level-rank duality.
