Integral Geometry of Plane Curvesand Knot

We study the integral expression of a knot invariant obtained as the second coeecient 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 invariants of generic plane curves recently deened 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. 1. Introduction The rst and second order terms in the perturbative expansion of Witten's Chern-Simons path integral associated with a knot in the 3-space were rst analyzed by Guadagnini, Martellini and Mintchev GMM] as well as by Bar-Natan BN] shortly after Witten's seminal work. In an announcement which appeared in 1992, Kontse-vich perceived a construction of a vast family of knot invariants which, presumably, contains the same information as the family of coeecients in the perturbative expansion of the Chern-Simons path integral associated with a knot K]. In a recent paper BT], Bott and Taubes explored this construction in a much more detailed manner. At this stage, it seems that a rigorous foundation has been laid for studying the per-turbative expansion of the Chern-Simons path integral associated with a knot. But, as it seems to us, we still lack a study of each individual knot invariant in this family in a way as concrete and thorough as possible. The rst term in the perturbative expansion turns out to be a classical quantity associated with a space curve with nowhere vanishing curvature, which was studied extensively under the name of the CC alugg areanu-Pohl-White self-linking formula. Although the second term as a knot invariant is also classical, we nd that the approach suggested by the perturbative theory of the Chern-Simons path integral provides deep insight into some of its pre