In this thesis, we analyze the Khovanov homology and symplectic Khovanov homology of links in the presence of various internal and external link symmetries. We define a new geometrically motivated annular theory for symplectic Khovanov homology, and apply localization techniques in Floer theory to produce spectral sequences relating the symplectic Khovanov homologies of 2-periodic and strongly invertible knots to the symplectic annular Khovanov homologies of their respective quotient annular knots. We also relate the symplectic annular Khovanov homology of a knot to the link Floerhomology of the lift of the annular axis in the double branched cover. Inspired by structural results for the symplectic Khovanov homology of strongly invertible knots, we also prove that the E-infinity page of the Lipshitz-Sarkar spectral sequence decomposes as the tensor product of the homology of a ‘reduced’ complex with H* (S¹).
Sriram Raghunath (Thu,) studied this question.