Exact Results in Quantum Theory

We discuss the imposition of the volume simplicity constraint in theEPRL spin foam model for quantum gravity, in particular, itsgeneralization to arbitrary vertices proposed by Kaminski, Kisielowskiand Lewandowski. As it turns out, this constraint is closely related tothe question of reconstruction of four-dimensional polytopes from theirface bivectors. We show how this can, in some cases, be formulated interms of knotting invariants of the boundary graphs, and discuss itsconnection to the non-geometricity modes of the model recently observedby some authors.