LATTICE KNOT-THEORY AND QUANTUM-GRAVITY IN THE LOOP REPRESENTATION

We present an implementation of the loop representation of quantum gra vity on a square lattice. Instead of starting from a classical lattice theory, quantizing, and introducing loops, we proceed backwards, sett ing up constraints in the lattice loop representation and showing that they have appropriate (singular) continuum limits and algebras. The d iffeomorphism constraint reproduces the classical algebra in the conti nuum and has as solutions lattice analogues of usual knot invariants. We discuss some of the invariants stemming from Chern-Simons theory in the lattice context, including the issue of framing. We also present a regularization of the Hamiltonian constraint. We show that two knot invariants from Chem-Simons theory are annihilated by the Hamiltonian constraint through the use of their skein relations, including interse ctions. We also discuss the issue of intersections with kinks. This pa per is the first step towards setting up the loop representation in a rigorous, computable setting.