A spin network generalization of the Jones polynomial and Vassiliev invariants

Abstract

We apply the ideas of Alvarez and Labastida to the invariant of spin networks defined by Witten and Martin based on Chern-Simons theory. We show that it is possible to define ambient invariants of spin networks that (for the case of SU (2)) can be considered as extensions to spin networks of the Jones polynomial. Expansions of the coefficients of the polynomial yield primitive Vassiliev invariants. The resulting invariants are candidates for solutions of the Wheeler-DeWitt equations in the spin network representation of quantum gravity.CGPG-97/10-4 q-alg/9711014Witten [1] showed that the expectation value of a Wilson loop in an SU (2) Chern-Simons theory,is a knot invariant closely related with the Jones polynomial. More precisely, it is given by the Kauffman bracket [2] knot polynomial. The latter differs from the Jones polynomial by an overall "phase factor" given by the exponential of the writhe of the knot. This factor accounts for the fact that the Kauffman bracket is a regular isotopy invariant (it is not invariant when one removes twists, i.e., it is an invariant of framed knots) whereas the Jones polynomial is an ambient isotopic invariant. The explicit relation between the Kauffman bracket and the expectation value we introduced above is given by K(γ, q) =< W (γ) > with q = exp(2πi/k), where q is the variable of the polynomial and k is the coupling constant of the Chern-Simons theory 1 . The expression of the invariants one obtains from the path integral appear as power series expansions in terms of the parameter 2πi/k. It is tantamount to evaluating the polynomials for q = exp(2πi/k) and expanding the exponential in power series. If one does this for the Jones polynomial, the resulting coefficients are Vassiliev invariants [5][6][7].In a separate context, it was also noticed that the resulting Vassiliev invariants are candidates for wavefunctions in the loop representation of quantum gravity [8][9][10][11]. In this context, however, one is not interested in arbitrary knot invariants, but actually in those that arise as counterparts of states in the "connection representation",This poses a series of restrictions on the possible knot invariants ψ(γ) that one could consider as wavefunctions of a quantum theory in the loop representation. These restrictions are known as "Mandelstam identities". If one concentrates of wavefunctions depending on a single loop, they are linear, loop dependent, relations i c i ψ(γ i ) = 0. They arise from the fact that the Wilson loop W (γ, A) is a trace of a group element [12]. Of all the Vassiliev invariants that one constructs using the techniques described above, we only know for a fact that two of them are compatible with these relations. This makes it particularly difficult to construct candidates for quantum states of the gravitational field in terms of these invariants. Implicit in this discussion is the fact that in quantum gravity one is interested in loops with generic intersections, otherwise the spectra of various quantum operators is not appropriately realized [1...

Canonical quantum gravity in the Vassiliev invariants arena: I. Kinematical structure

“…Their calculations were based on the use of monodromies and the tangle group, respectively. For trivalent vertices we have also shown in a previous paper [25] that one can extract the framing factor and construct ambient isotopic invariants.…”

“…The four-valent case presents considerable technical challenges, since most of the literature on spin network invariants has concentrated heavily on trivalent vertices [21,22,24]. In particular, the whole issue of constructing framing-independent spin network invariants starting from Chern-Simons theory has received virtually no attention (see [25] for first attempts). Note that, in principle, these are the invariants of most interest for the gravitational case, where wavefunctions are supposed to be invariant under diffeomorphisms.…”

“…In the above expressions, we have adopted the usual 'generalized Einstein convention' of extended loops, in which a summation is implied by any repeated index a i and an integral over the 3-manifold is implied by any repeated index x i . Some general results concerning the factorization of the geometric factors (see equation (25)) can be derived from the following property of these fields:…”

Embedded graph invariants in Chern-Simons theory

“…Recently Gambini, Griego, and Pullin, building on earlier work by Alvarez and Labastida [22], have proposed that Vassiliev invariants, generalized to include spin net states, are solutions to the Hamiltonian constraint of quantum gravity [29]. This construction is based on the idea that the framing dependence may be collected into an overall factor in the expectation value of Wilson loops.…”