String Math Seminar: Cheuk Yu Mak (University of Sheffield) “symplectic annular Khovanov homology and fixed point localisation”

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Abstract:

Khovanov homology is a powerful link invariant which has numerous applications. It is powerful not only because it is a strong invariant, but also it is functorial and has many relations with other invariants. In 2018, Stoffregen-Zhang realized that there is a spectral sequence from the Khovanov homology of a periodic link to the annular Khovanov homology of its quotient, which was later used by Baldwin-Hu-Sivek to show that Khovanov homology detects the (2,5) torus knot (i.e. any knot whose reduced Khovanov homology is the same as that of the (2,5) torus knot is the (2,5) torus knot). In 2022, Lipshitz-Sarkar extends Stoffregen-Zhang’s result to strongly invertible knot and the infinity page of the spectral sequence is, mysteriously, the cone of the annular Khovanov homology of the axis moving map between the two resolutions of the quotient.

In this talk, we will explain the symplectic analogues of these spectral sequences and why they are easy consequences of fixed point localisation theorem of Lagrangian Floer theory. This is a joint work with Hendricks and Raghunath.