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 Σ 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 D(Σ)^* of Σ as being the Schreier coset graph of the arrow presentation, explains the ribbon structure behind curves on D(Σ)^* 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.