Gauge Theory and Topology: Fillable contact structures from positive surgery

Gauge Theory and Topology Seminar, SEMINARS

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March 10, 2023 10:30 am - 11:30 am
Science Center 507
Address: 1 Oxford Street, Cambridge, MA 02138 USA

Thomas Mark - University of Virginia

For a Legendrian knot $K$ in a closed contact 3-manifold, we describe a necessary and sufficient condition for contact $n$-surgery along $K$ to yield a weakly symplectically fillable contact manifold, for some integer $n>0$. When specialized to knots in the standard 3-sphere this gives an effective criterion for the existence of a fillable positive surgery, along with various obstructions. These are sufficient to determine, for example, whether such a surgery exists for all knots of up to 10 crossings. The result also has certain purely topological consequences, such as the fact that a knot admitting a lens space surgery must have slice genus equal to its 4-dimensional clasp number. We will mainly explore these topologically-flavored aspects, but will give some hints of the general proof if time allows.