A spectral sequence from Khovanov homology to knot Floer homology (Q6579949)

This paper constructs the long-sought spectral sequence from reduced Khovanov homology to the reduced knot Floer homology. The positive resolution of a 20 year old conjecture of \textit{J. Rasmussen} [Fields Inst. Commun. 47, 261--280 (2005; Zbl 1095.57016)] follows as a corollary. The author suggests that this spectral sequence is the Heegaard Floer analogue of the Kronheimer-Mrowka spectral sequence that abuts to instanton Floer homology [\textit{P. B. Kronheimer} and \textit{T. S. Mrowka}, Publ. Math., Inst. Hautes Étud. Sci. 113, 97--208 (2011; Zbl 1241.57017)].\N\NThe spectral sequence of this paper is defined combinatorially, via a homology theory constructed in previous work of the author [\textit{N. Dowlin}, ``A family of $\mathfrak{sl}_{n}$-like invariants in knot Floer homology'', Preprint, \url{arXiv:1804.03165}]. In Theorem 1.4 it is established that the differential on the \(k\)-th page of the spectral sequence is of bidegree \((2k+3, 4k+3)\). This allows for the computation of this sequence for cases in which the Kronheimer-Mrowka sequence is currently unknown e.g. \(T(4,5)\) (but see [\textit{A. Lobb} and \textit{R. Zentner}, Algebr. Geom. Topol. 20, No. 2, 531--564 (2020; Zbl 1508.57017)]).\N\NTheorem 7.8 provides evidence that the construction of this spectral sequence may extend to the minus theory of knot Floer homology.