Much of condensed matter physics is concerned with understanding how different kinds of order emerge from interactions between a large number of simple constituent. In ordered phase such as crystals, magnets, and superfluids, the ordering is characterised through the symmetry breaking mechanism in terms of a local order parameter. Starting around 1982 new mathematical ideas of quantum geometry and topology ideas have penetrated into condensed matter theory and led to new insights and way s of looking at condensed matter. Recently they led to the discovery of the topological insulators, a new exciting class of materials that shocked material scientists who had studied these materials without noticing these properties. Geometric order is preserved in Topological insulators. This order is a result of the time reversal symmetry. In this thesis one of such models are studied. In the mathematical expression knotted forms have nontrivial topology and knotless forms have trivial topology. In quantum mechanics, the topology of a quantum state does not change, as long as there is no energy gap. Time reversal invariance is conserved in gapless nodes and kramers degeneration occurs in these points. Time reversal symmetry is one of the most important symmetries in nature. The interior side of a topological insulators is gapped, while the gapless edge state of these materials cause conduction at the boundaries. In this thesis we introduce two models in one and two dimensions and investigate its topological characteristics by calculating the topological indices. One of the wellknown indices is the chern number. In $1998$, Haldane proposed a model that had a non-trivial topology. In the Haldane model, there is a quantum Hall effect is happened without magnetic field and time reversal symmetry breaking. The model which is proposed by Haldane is known as a chern insulator. Topological index in one dimension is called zak phase, Which is also determined experimentally. For this purpose a diatomic chain of rubidium is investigated, which is described by the Rice-Mele Hamiltonian. We study dependance of topological invariant to the hopping integrals. We also determine the condition of existance of the edge states and obtaine.
