FILLABLE CONTACT STRUCTURES FROM POSITIVE SURGERY.

We consider the question of when the operation of contact surgery with positive surgery coefficient, along a knot K in a contact 3-manifold Y, gives rise to a weakly fillable contact structure. We show that this happens if and only if Y itself is weakly fillable, and K is isotopic to the boundary of a properly embedded symplectic disk inside a filling of Y. Moreover, if Y - is a contact manifold arising from positive contact surgery along K, then any filling of Y - is symplectomorphic to the complement of a suitable neighborhood of such a disk in a filling of Y . Using this result we deduce several necessary conditions for a knot in the standard 3-sphere to admit a fillable positive surgery, such as quasipositivity and equality between the slice genus and the 4-dimensional clasp number, and we give a characterization of such knots in terms of a quasipositive braid expression. We show that knots arising as the closure of a positive braid always admit a fillable positive surgery, as do knots that have lens space surgeries, and suitable satellites of such knots. In fact the majority of quasipositive knots with up to 10 crossings admit a fillable positive surgery. On the other hand, in general, (strong) quasipositivity, positivity, or Lagrangian fillability need not imply a knot admits a fillable positive contact surgery. [ABSTRACT FROM AUTHOR]