Knots, minimal surfaces and J-holomorphic curves

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• Joel Fine, Université Libre de Bruxelles
• Wednesday 26 January 2022, 16:00-17:00
• MR13.

If you have a question about this talk, please contact Henry Wilton.

I will describe work in progress, parts of which are joint with Marcelo Alves. Let K be a knot or link in the 3-sphere. I will explain how one can count minimal surfaces in hyperbolic 4-space which have ideal boundary equal to K, and in this way obtain a link invariant. In other words the number of minimal surfaces doesn’t depend on the isotopy class of the link. These counts of minimal surfaces can be organised into a two-variable polynomial which is perhaps a known polynomial invariant of the link, such as HOMFLYPT .

“Counting minimal surfaces” needs to be interpreted carefully here, similar to how Gromov-Witten invariants “count” J-holomorphic curves. Indeed I will explain how these minimal surface invariants can be seen as Gromov-Witten invariants for the twistor space of hyperbolic 4-space. This leads naturally to a new class of infinite-volume 6-dimensional symplectic manifolds with well behaved counts of J-holomorphic curves. This gives more potential knot invariants, for knots in 3-manifolds other than the 3-sphere. It also enables the counting of minimal surfaces in more general Riemannian 4-manifolds, besides hyperbolic space.

This talk is part of the Differential Geometry and Topology Seminar series.

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