Knots and quivers, HOMFLY and DT

Physicists have long been arguing that gauge theories at large rank should be related to topological string theories. To give a concrete example of such a relation, I will describe a new conjectural correspondence between the colored HOMFLY-PT polynomials of knots and the motivic DT invariants of certain symmetric quivers. I will outline a proof of this conjecture for rational knots and then speculate about how much of the HOMFLY-PT skein theory might carry over to the realm of DT quiver invariants. This is joint work with Marko Stosic.