Slice alternating knots, rational balls, and lattice embeddings

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• Brendan Owens (Glasgow)
• Wednesday 17 May 2023, 16:00-17:00
• MR13.

If you have a question about this talk, please contact Oscar Randal-Williams.

A fundamental problem in smooth 4-dimensional topology is to understand which surfaces in the 4-ball can be bounded by a given classical knot or link, and in particular, whether a given knot is slice (bounds a disk). A related problem is to understand when a given 3-manifold bounds a rational homology 4-ball. I will introduce these problems and then focus on the case of alternating knots and links, and describe some recent work in two directions: 1) the determination of the sliceness of 99.9997% of the over 1.2 billion prime alternating knots with up to 21 crossings, joint with Frank Swenton, and 2) progress towards a conjectured characterisation of when the double branched cover of an alternating link bounds a rational ball, joint with Josh Greene, and using work of Greene and Jabuka. The key tool is Donaldson’s diagonalisation theorem, augmented with some Heegaard Floer theory.

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

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