Symplectic fillings, contact surgeries, and Lagrangian disks

TL;DR

This paper characterizes when contact (r)-surgery on Legendrian knots produces symplectically fillable manifolds, provides obstructions for certain surgeries, and explores Lagrangian fillings of Legendrian knots.

Contribution

It completely determines the fillability conditions for contact (r)-surgery on Legendrian knots in the 3-sphere.

Findings

Contact (r)-surgery yields fillable manifolds for specific r in (0,1]

Obstructions are identified for positive r outside this range

Lagrangian fillings of Legendrian knots are investigated

Abstract

This paper completely answers the question of when contact (r)-surgery on a Legendrian knot in the standard contact structure on the 3-sphere yields a symplectically fillable contact manifold for r in (0,1]. We also give obstructions for other positive r and investigate Lagrangian fillings of Legendrian knots.