  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.
