Mathematics Colloquia and Seminars

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An important problem in smooth topology is to understand the 4-ball genus and the 4-ball crossing number of a smooth knot. More generally, one can study the “geography” question of which combinations of genus and transverse double points can be realized by a smooth, orientable, immersed surface in the 4-ball that has the given knot as its boundary. I will discuss analogous problems when the knot and surface satisfy additional geometric conditions imposed by symplectic geometry: the surface is Lagrangian and the boundary knot is Legendrian. Whereas any smooth knot can be filled by an infinite number of topologically distinct embedded surfaces, there are classical and non-classical obstructions to the existence of embedded Lagrangian fillings of Legendrian knots. Legendrian knots that admit a generating family will always admit “compatible” immersed Lagrangian fillings, and I will describe how a generating family polynomial can obstruct the existence of some types of immersed fillings. I will also describe some constructions of immersed fillings. We will see that there are Legendrian knots for which the polynomial can completely capture all types of immersed fillings that can be realized, but for other Legendrian knots there are deeper algebraic obstructions. Most of this work is joint with Samantha Pezzimenti, but I will also briefly mention some joint work with Orsola Capovilla-Searle, Noémie Legout, Maÿlis Limouzineau, Emmy Murphy, and Yu Pan.