The search for solutions was continued by Ezawa  (see also  for an extensive review), who used a symmetrized form of the Hamiltonian. His solutions depend on multiple plaquette loops , where a single lattice plaquette is traversed by the loop times. The solutions are less trivial than those formed from Polyakov loops, since they involve kinks, but they are still annihilated by the volume operator.
A somewhat different strategy was followed by Fort et al. , who constructed a Hamiltonian lattice regularization for the calculation of certain knot invariants. They defined lattice constraint operators in terms of their geometric action on lattice Wilson loop states, and reproduced some of the formal continuum solutions to the polynomial Hamiltonian constraint of complex Ashtekar gravity on simple loop geometries.
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