Transverse surgery on knots in contact three-manifolds

Abstract

We study the effect of surgery on transverse knots in contact $3$-manifolds by examining its effect on open books, the Heegaard Floer contact invariant, and tightness in general. We first compare surgery on transverse knots to that on Legendrian knots. We then show how to give an open book decomposition for most admissible and all inadmissible transverse surgeries on binding components of open books. We show that if we perform inadmissible transverse r-surgery on the connected binding of a genus g open book that supports a tight contact structure, then this operation preserves tightness if the surgery coefficient r is greater than 2g-1. We also give criteria for when positive contact surgery on Legendrian knots will result in an overtwisted manifold. We explicitly show that all positive contact surgeries on Legendrian figure-eight knots in S^3 with its standard tight contact structure result in overtwisted contact manifolds.