Algebraic overtwisted contact structures on 3-sphere

It is known that all of the complex analytic singularity links and the associated Milnor open books on the 3-sphere correspond to a single contact structure, which is the unique tight structure of the 3-sphere. The main question of this thesis is whether the overtwisted contact structures on the 3-sphere are real algebraic. We will define the notion of real algebraicity in the introduction of the thesis. We explicitly construct a family of real algebraic multilinks in the 3-sphere which are the bindings of planar Milnor open book decompositions supporting overtwisted contact structures. Furthermore, we prove that all the overtwisted contact structures with non-negative 3-dimensional invariants are obtained in this family.