We iterate Manolescu's unoriented skein exact triangle in knot Floer homology
with coefficients in the field of rational functions over
$\mathbb{Z}/2\mathbb{Z}$. The result is a spectral sequence which converges to
a stabilized version of delta-graded knot Floer homology. The $(E_2,d_2)$ page
of this spectral sequence is an algorithmically computable chain complex
expressed in terms of spanning trees, and we show that there are no higher
differentials. This gives the first combinatorial spanning tree model for knot
Floer homology.Comment: 58 pages, 18 figures. Published version, with updated reference
We exhibit a knot P in the solid torus, representing a generator of first homology, such that for any knot K in the 3-sphere, the satellite knot with pattern P and companion K is not smoothly slice in any homology 4-ball. As a consequence, we obtain a knot in a homology 3-sphere that does not bound a piecewise-linear disk in any homology 4-ball.
Abstract. We show that the integer homology sphere obtained by splicing two nontrivial knot complements in integer homology sphere L-spaces has Heegaard Floer homology of rank strictly greater than one. In particular, splicing the complements of nontrivial knots in the 3-sphere never produces an L-space. The proof uses bordered Floer homology.
Abstract. A well-known conjecture states that for any l-component link L in S 3 , the rank of the knot Floer homology of L (over any field) is less than or equal to 2 l−1 times the rank of the reduced Khovanov homology of L. In this paper, we describe a framework that might be used to prove this conjecture. We construct a modified version of Khovanov homology for links with multiple basepoints and show that it mimics the behavior of knot Floer homology. We also introduce a new spectral sequence converging to knot Floer homology whose E 1 page is conjecturally isomorphic to our new version of Khovanov homology; this would prove that the conjecture stated above holds over the field Z 2 .
Let K be a rationally null-homologous knot in a 3-manifold Y , equipped with a nonzero framing λ, and let Y λ (K) denote the result of λ-framed surgery on Y . Ozsváth and Szabó gave a formula for the Heegaard Floer homology groups of Y λ (K) in terms of the knot Floer complex of (Y, K). We strengthen this formula by adding a second filtration that computes the knot Floer complex of the dual knot K λ in Y λ , i.e., the core circle of the surgery solid torus. In the course of proving our refinement we derive a combinatorial formula for the Alexander grading which may be of independent interest.
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