Khovanov Homology and the Involutive Heegaard Floer Homology of Branched Double Covers

Involutive Heegaard Floer homology (HFI), introduced by Hendricks and Manolescu in 2017, incorporates the conjugation action on Ozsváth and Szabó's Heegaard Floer homology to produce a richer 3-manifold invariant. In 2019, Hendricks and Lipshitz introduced Involutive Bordered Floer homology, a modular (i.e. cut-and-paste) version of HFI, and also exhibited a surgery exact triangle for the theory.

In joint work with Akram Alishahi and Linh Truong, we develop an involutive extension of Lipshitz, Ozsváth, and Thurston's bordered construction of Ozsváth and Szabó's spectral sequence from the Khovanov homology of (the mirror of) a knot to the Heegaard Floer homology of its branched double cover. This spectral sequence can be compared with Lin's spectral sequence from a version of Khovanov homology to the monopole Floer homology of the branched double cover.

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