VARIATIONAL DERIVATION OF EXACT SKEIN RELATIONS FROM CHERN-SIMONS THEORIES

The expectation value of a Wilson loop in a Chern-Simons theory is a k not invariant. Its skein relations have been derived in a variety of w ays, including variational methods in which small deformations of the loop are made and the changes evaluated. The latter method is only all owed to obtain approximate expressions for the skein relations. We pre sent a generalization of this idea that allows to compute the exact fo rm of the skein relations. Moreover, it requires to generalize the res ulting knot invariants to intersecting knots and links in a manner con sistent with the Mandelstam identities satisfied by the Wilson loops. This allows for the first time to derive the full expression for knot invariants that are suitable candidates for quantum states of gravity (and supergravity) in the loop representation. The new approach leads to several new insights in intersecting knot theory, in particular the role of non-planar intersections and intersections with kinks.