Link invariants for intersecting loops

Abstract

We generalize the braid algebra to the case of loops with intersections. We introduce the Reidemeister moves for four-and six-valent vertices to have a theory of rigid vertex equivalence. By considering representations 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 polynomials for intersecting links correspond to the vacuum expectation value of the Wilson line operator of the Chern-Simons theory. We make contact with quantum gravity by showing that these polynomials are simply related with some solutions of the complete set of constraints with cosmological constant Λ, for loops including triple self intersections. Previous derivations of this result were restricted to the four-valent case.