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The spin network quantum simulator relies on the su(2) representation ring(or its q-deformed counterpart at q= root of unity) and its basic featuresnaturally include (multipartite) entanglement and braiding. In particular,q-deformed spin network automata are able to perform efficiently approximatecalculations of topological invarians of knots and 3-manifolds. The samealgebraic background is shared by 2D lattice models supporting topologicalphases of matter that have recently gained much interest in condensed matterphysics. These developments are motivated by the possibility to store quantuminformation fault-tolerantly in a physical system supporting fractionalstatistics since a part of the associated Hilbert space is insensitive to localperturbations. Most of currently addressed approaches are framed within a'double' quantum Chern-Simons field theory, whose quantum amplitudes representevolution histories of local lattice degrees of freedom.  We propose here a novel combinatorial approach based on `state sum' models ofthe Turaev-Viro type associated with SU(2)_q-colored triangulations of theambient 3-manifolds. We argue that boundary 2D lattice models (as well asobservables in the form of colored graphs satisfying braiding relations) couldbe consistently addressed. This is supported by the proof that the Hamiltonianof the Levin-Wen condensed string net model in a surface Sigma coincides withthe corresponding Turaev-Viro amplitude on Sigma x [0,1] presented in the lastsection.

BRAIDING AND ENTANGLEMENT IN SPIN NETWORKS: A COMBINATORIAL APPROACH TO TOPOLOGICAL PHASES