# String-net models on non-trivalent lattices

I have just started reading about string net models. The following aspect wasn't entirely clear to me:

String net models are most naturally defined on trivalent networks, that is to say networks where we attach 3 "legs" to each "point" of the lattice. The model is then fully specified by defining the string types, branching rules (the "$N_{ij}^{k}$") and the orientation of the strings.

While this approach appears natural (especially based on what I have read on Tensor categories) most of the "simplest" String net models (such as the $Z_2$ Kitaevs Toric Code or more generically the $Z_N$ Wen Plaquette Model) can also be defined on square lattices where we seem to have 4 "legs" attached to each "point".

I was wondering how and whether one can always "reduce" a String net model on an arbitrary N-valent lattice (N "legs" attached to each "point") to a String Net Model on a trivalent lattice. Similarly: Can one construct arbitrary String Net Models on N-Valent lattices by understanding the model on some trivalent lattice?

But in order not to complicate things: What is the "recipe" to reduce the Z2 Kitaev Toric Code on a planar square lattice to some "trivalent lattice"?

I am looking forward to your responses!