A plane field ξ on a 3-manifold M is called a contact structure if there exists a 1-form α such that ξ=kerpαq and α ^ dα \=0. Two contact structures ξ0 and ξ1 on M are contactomorphic if there exists a diffeomorphism f : M Ñ M such that f˚pξ0q “ ξ1. Jean Martinet showed that any oriented closed 3-manifold admits a contact structure [JM]. A contact manifold is overtwisted if it contains an embedded disk where the contact structure is tangent to the disk on the boundary. A contact structure that is not overtwisted is called tight.

A knot in a contact manifold is called Legendrian if it is tangent to the con- tact plane at each point. There is a way to construct tight contact structures from overtwisted contact structures. We look at a special class of knots called non-loose knots in overtwisted contact structures. A knot in overtwisted contact structures is called non-loose if it intersects all the overtwisted disks transversely. Therefore, the complement of a non-loose knot is tight. We would like to understand how tight complements behave under the operations of connected sum and Whitehead doubling in overtwisted contact structures. Currently, we are working on understanding the connect sum operation and we have made some progress.

Another problem we would like to work on is checking if we get contactomorphic contact structures after ´1 contact surgery on two non-isotopic Legendrian knots with the same classical invariants. The set of knots under consideration here are non-simple Legendrian twist knots that Etnyre, NG, and V´ertesi classified. Since the knots are not isotopic we knot that the complements are not contactomorphic. A better understanding of the complement will help us understand the contact structure one gets after surgery.