Khovanov homology, instantons, and link detection

Khovanov homology, instantons, and link detection

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John Baldwin, Boston College
Fine Hall 314

In 2010, Kronheimer and Mrowka proved that Khovanov homology detects the unknot, answering a categorical version of the famous open question of whether the Jones polynomial detects the unknot. Their proof makes use of a spectral sequence relating Khovanov homology with a version of instanton Floer homology for links. Last year, Steven Sivek and I used their spectral sequence together with ideas in sutured manifold theory and contact geometry to prove that Khovanov homology also detects the right- and left-handed trefoils. I'll discuss this result and some of the key elements of its proof. I'll end with some open questions related to link detection (does Khovanov homology detect the Hopf link?) and knot surgery (are knots with SU(2)-abelian surgeries fibered with 3-genus equal to smooth 4-genus?) which we hope to answer in the near future.