Integral Geometry of Plane Curves and Knot Invariants

We study the integral expression of a knot invariant obtained as the second coefficient in the perturbative expansion of Witten's Chern-Simons path integral associated with a knot. One of the integrals involved turns out to be a generalization of the classical Crofton integral on convex plane curves, and it is related with the invariants of generic plane curves recently defined by Arnold, with deep motivations in symplectic and contact geometry. Quadratic bounds on these plane curve invariants are derived using their relationship with the knot invariant.