BIRS Workshop Lecture Videos
The earring correspondence on the pillowcase Kotelskiy, Artem
Given a decomposition of a knot K into two four-ended tangles T and T', the (holonomy perturbed) traceless-SU(2)-character-variety functor produces Lagrangians R(T) and R(T') in the pillowcase P. Hedden, Herald and Kirk used this to define Pillowcase homology, conjecturally the symplectic counter-part of the singular instanton homology I(K). Important in their construction is how R(T) and its restriction to P are affected by â adding an earringâ , a process used by Kronheimer and Mrowka to avoid reducibles. The object that governs this process turns out to be an immersed Lagrangian correspondence from pillowcase to itself. We will describe this correspondence in detail, and study its action on Lagrangians. In the case of the (4,5) torus knot, we will see that a correction term from the bounding cochains must be added. We will indicate a particular figure eight bubble which recovers the desired bounding cochain, as predicted by Bottman and Wehrheim. This is ioint work with G. Cazassus, C. Herald and P. Kirk.
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