The Evaluation of Graphs on Surfaces for State-Sum Models with Defects

The evaluation of graphs on 2-spheres is a central ingredient of the Turaev–Viro construction of three-dimensional topological field theories. In this article, we introduce a class of graphs, called extruded graphs, that is relevant for the Turaev–Viro construction with general defect configurations involving defects of various dimensions. We define the evaluation of extruded graphs and show that it is invariant under a set of moves. This ensures the computability and uniqueness of our evaluation.