  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.
