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Fractional Dehn twists in knot theory and contact topology

Fractional Dehn twists give a measure of the difference between the relative isotopy class of a homeomorphism of a bordered surface and the Thurston representative of its free isotopy class. We show how to estimate and compute these invariants. We discuss the the relationship of our work to stabilization problems in classical knot theory, general open book decompositions, and contact topology. We include an elementary characterization of overtwistedness for contact structures described by open book decompositions.