The colored Jones polynomial is among the most important quantum knot invariants and is conjectured—via the volume conjecture—to encode rich geometric information, such as hyperbolic volume and Chern–Simons invariants and Reidemeister torsion.

In this talk, we present explicit computations of the colored Jones polynomials for plat closures of 4-strand braids using purely diagrammatic methods within the framework of relative Kauffman bracket skein modules.

Though we use entirely combinatorial in nature, this approach is closely related to the representation theory of braid groups acting on quantum planes via an unoriented version of the Reshetikhin–Turaev construction.
This connection, in turn, allows us to interpret the construction in terms of q-analogues of Plücker coordinates on the Grassmannian Gr(2,4).