Seminar of Algebra

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Every knot with unknotting number $< 22$ bounds a smooth disc in a punctured $K_3$ surface

Marco Marengon (Alfréd Rényi Institute of Mathematics)
Departamento de Álgebra
Mon, 26 jun 2023 12:00
My research area is low-dimensional topology, including knot concordance, exotic manifolds, Heegaard Floer homology, and Khovanov homology.

A question in knot theory that has recently become popular is to classify what knots bound a smooth disc in $X - \mathrm{int}(B^4)$, where $X$ is a given closed 4-manifold. We study the case when $X$ is the $K3$ surface, and prove that every knot with unknotting number less than 22 bounds a smooth disc in $K3 - \mathrm{int}(B^4)$. Our proof is constructive and based on the existence of a plumbing tree of 22 spheres in $K3$. This is joint work with Stefan Mihajlović.