Abstract:
Singular instanton homology is a knot invariant introduced by Kronheimer and Mrowka. It is deeply tied to Khovanov homology, and among other things, permits to show that the latter detects the unknot.In order to compute singular instanton homology, Hedden, Herald and Kirk defined a symplectic (Atiyah-Floer) analogue, called pillowcase homology. This is a Lagrangian Floer homology in the traceless character variety of the four-punctured sphere.
We study the Lagrangian correspondence induced by the earring tangle, an essential ingredient in Kronheimer-Mrowka's construction. Our computation suggests that figure eight bubbling — a subtle degeneration phenomenon predicted by Bottman and Wehrheim — appears in the context of traceless character varieties. This is joint work with Chris Herald, Paul Kirk and Artem Kotelskiy.