Yanbai Zhang's 2017 honours thesis in mathematics, on "From the Temperley-Lieb Categories to Toric Code", supervised by

Scott Morrison, is available here.

### Outline

In this thesis our aim is to show the ground states of Hamiltonians on 2-dimensional surfaces, which is made up by putting together stablizer operators corresponding to vertices, edges and faces, coincide with topology of surface. We first introduce the Temperley-Lieb categories and then define the skein module of $TL(\delta = 1)$, which is the specialization of the Temperley-Lieb category at $\delta = 1$. Then we introduce the Levin-Wen models, which is the generalization of the toric code, with general defined projections (operators in quantum physics) corresponding to edges and vertices. We then prove the kernel of Hamiltonian in the toric code is isomorphic to the skein module of $TL(\delta = 1)$ and therefore only depends on the topology of surface it is based on.