A class of tight contact structures on the genus two surface cross the interval
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A contact structure » on a three manifoldM is a completely non-integrable tangent two-plane distribution. A contact structure is called overtwisted if it contains an embedded disk D such that » is everywhere tangent to D along @D. Eliashberg  showed that the topologically interesting case to study is tight contact structures. He did this by showing that the classes of overtwisted contact structures correspond to homotopy classes of two-plane distributions on M. The purpose of this work is to classify tight contact structures on M = §2 £I with a speci¯ed boundary condition. This is done by applying cut and paste contact topological techniques developed by Honda, Kazez and Mati¶c [15, 18].