Research Project

A Milnor fillable 3-manifold is defined as a closed, oriented, contact 3-manifold that is contact isomorphic to the contact link manifold of a complex analytic surface with isolated singularity. It is known that such manifolds admit a unique Milnor fillable contact structure up to contactomorphism. As the contact structures on 3-manifolds are of two types, called tight and overtwisted, it is known that the Milnor fillable contact structure of a Milnor fillable manifold is tight. A natural question one can ask is whether one can produce overtwisted contact structures of a Milnor fillable manifold in an analytical way. 

When we restrict ourselves to 3-spheres, there is unique tight contact structure and countable infinitely many overtwisted contact structures which are distinguished by the half integer valued d_3 invariant.  It is already proven that all overtwisted structures in 3-spheres are real algebraic by explicit construction. However, the constructed open book decompositions that supports the contact structures has pages with varying genus. In our previous work, by considering Seifert/graph multilinks, we have shown that every overtwisted contact structure on 3-spheres with positive d_3 invariant is real algebraic and the associated open books are planar (except 13 of them). As it is known that any overtwisted contact structure is supported by a planar open books, in this project, we will investigate whether there is a real algebraic planar overtwisted contact structures on 3-spheres with negative d_3 invariant.  In our previous work, while constructing wider families of graph multilinks, we used splicing operation, which is a topological operation preserving the algebraicity of the multilinks. A further question we are planning to consider is to find an algebraic paste operation of 4-manifolds which corresponds to the splicing operation on the boundary of the manifold. 

Duration of stay: 1st February to 1st June 2023