Fillable contact structures from positive surgery

TL;DR

This paper characterizes when positive contact surgery on a knot yields a fillable contact structure, linking fillability to properties of the knot and the ambient manifold, and identifies classes of knots that admit such surgeries.

Contribution

It provides a complete characterization of fillable positive contact surgeries in terms of the knot and the ambient manifold, and relates fillability to quasipositivity and braid representations.

Findings

Fillable positive surgery occurs iff the manifold is weakly fillable and the knot bounds a symplectic disk.

Knots with fillable positive surgery include closures of positive braids and knots with lens space surgeries.

Most quasipositive knots with up to 10 crossings admit fillable positive surgery.

Abstract

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.