This dissertation contains two results. The first result involves concave symplectic struc-tures on a neighborhood of certain plumbing of symplectic surfaces, introduced by D. Gay.We draw the contact surgery diagram of the induced contact structure on boundary of aconcave filling, when the induced open book is planar. We show that every Brieskorn sphereadmits a concave Filling in the sense of D. Gay and the induced contact structure on it isovertwisted. We also show that in certain cases a (-1)-sphere in Gay's plumbing can beblown down to obtain a concave plumbing of the same type. The next result examines thecontact structure induced on the boundary of the cork W1, induced by the double branchedcover over a ribbon knot. We show this contact structure is overtwisted in a specific case.
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