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A well-known conjecture of Rasmussen states that for any knot K in S^{3}, the rank of the reduced Khovanov homology of K is greater than or equal to the rank of the reduced knot Floer homology of K. This rank inequality is supposed to arise as the result of a spectral sequence from Khovanov homology to knot Floer homology. Using an oriented cube of resolutions construction for a homology theory related to knot Floer homology, we prove this conjecture. [ABSTRACT FROM AUTHOR]

In this brief note, we investigate the \mathbb {CP}^2-genus of knots, i.e., the least genus of a smooth, compact, orientable surface in \mathbb {CP}^2\smallsetminus \mathring {B^4} bounded by a knot in S^3. We show that this quantity is unbounded, unlike its topological counterpart. We also investigate the \mathbb {CP}^2-genus of torus knots. We apply these results to improve the minimal genus bound for some homology classes in \mathbb {CP}^2\# \mathbb {CP} ^2. [ABSTRACT FROM AUTHOR]

Secondary homological stability is a recently discovered stability pattern for a sequence of spaces exhibiting homological stability and it holds outside the range where the homology stabilizes. We prove secondary homological stability for the homology of the unordered configuration spaces of a connected manifold. The main difficulty is the case that the manifold is closed because there are no obvious maps inducing stability and the homology eventually is periodic instead of stable. We resolve this issue by constructing a new chain-level stabilization map for configuration spaces. [ABSTRACT FROM AUTHOR]