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.

Let C Z denote the group of knots in homology spheres that bound homology balls, modulo smooth concordance in homology cobordisms. Answering a question of Matsumoto, the second author previously showed that the natural map from the smooth knot concordance group C to C Z is not surjective. Using tools from Heegaard Floer homology, we show that the cokernel of this map, which can be understood as the non-locally-flat piecewise-linear concordance group, is infinitely generated and contains elements of infinite order.

We use grid diagrams to give a combinatorial algorithm for computing the knot Floer homology of the pullback of a knot K S 3 in its m-fold cyclic branched cover † m .K/, and we give computations when m D 2 for over fifty three-bridge knots with up to eleven crossings.

We construct infinitely many compact, smooth 4-manifolds which are homotopy equivalent to $S^{2}$ but do not admit a spine (that is, a piecewise linear embedding of $S^{2}$ that realizes the homotopy equivalence). This is the remaining case in the existence problem for codimension-2 spines in simply connected manifolds. The obstruction comes from the Heegaard Floer $d$ invariants.