Knot invariants and Oainleve equations

On the conference "Knots and Links in Fluid Flows" in IUM at one talk the following result was formulated. A HOMFLY invariant of a toric knot $(n,n+1)$ can be expressed through variables that describe a discrete dynamical system. Recurrent relations for these variables porvide that the generating function of these variables satisfy the Painlvev 2 equation.
I am going to tell about invariants of knots that come from four-dimensiona manifolds. Actually the invariants of manifolds can be organized into formal solutions of PDE. In particulat I'll explain how to a symplectic four-dimensional manifold there corresponds a solution of WDVV. And among reductions of these equations there are Painleve and Kdf equations