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Detecting the trefoil

Banff International Research Station for Mathematical Innovation and Discovery

I'll describe a simple proof (joint with Vela-Vick) that the rank of knot Floer homology detects the trefoil and that L-space knots are prime, results which were originally proven by Hedden-Watson and Krcatovich, respectively. Our argument is very Heegaard-diagram centric, but I'll describe an alternative proof which is more contact-geometric and uses Etnyre-Vela-Vick's "limit" description of knot Floer homology. The advantage of this geometric approach is that it can be (we think) ported to the instanton Floer setting to show that the rank of (sutured) instanton knot Floer homology detects the trefoil. If this all works it would, in combination with Kronheimer-Mrowka's spectral sequence relating Khovanov homology and singular instanton knot homology, prove that Khovanov homology detects the trefoil. The latter work is joint with Sivek.

Mathematics; Manifolds and cell complexes; Global analysis, analysis on manifolds; Low dimensional topology

video/mp4

Author affiliation: Boston College

2016-09-24

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/59283

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/