Oriented Closed Polyhedral Maps and the Kitaev Model

A kind of combinatorial map, called arrow presentation, is proposed to encode the data of the oriented closed polyhedral complexes $\Sigma$ on which the Hopf algebraic Kitaev model lives. We develop a theory of arrow presentations which underlines the role of the dual of the double $\mathcal{D}(\Sigma)^*$ of $\Sigma$ as being the Schreier coset graph of the arrow presentation, explains the ribbon structure behind curves on $\mathcal{D}(\Sigma)^*$ and facilitates computation of holonomy with values in the algebra of the Kitaev model. In this way, we can prove ribbon operator identities for arbitrary f.d. C$^*$-Hopf algebras and arbitrary oriented closed polyhedral maps. By means of a combinatorial notion of homotopy designed specially for ribbon curves, we can rigorously formulate “topological invariance” of states created by ribbon operators.