LINK INVARIANTS FOR INTERSECTING LOOPS

We generalize the braid algebra to the case of loops with intersection s. We introduce the Reidemeister moves for four- and six-valent vertic es to have a theory of rigid vertex equivalence. By considering repres entations of the extended braid algebra, we derive skein relations for link polynomials, which allow us to generalize any link polynomial to the intersecting case. We perturbatively show that the HOMFLY polynom ials for intersecting links correspond to the vacuum expectation value of the Wilson line operator of the Chem-Simons theory. We make contac t with quantum gravity by showing that these polynomials are simply re lated with some solutions of the complete set of constraints with cosm ological constant LAMBDA, for loops including triple self intersection s. Previous derivations of this result were restricted to the four-val ent case.