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Using Dowlin's spectral sequence from Khovanov homology to knot Floer homology, we prove that reduced Khovanov homology (over $\mathbb{Q}$) detects the figure-eight knot.

Using Heegaard Floer homology, we construct a numerical invariant for any smooth, oriented 4 4 -manifold X X with the homology of S 1 × S 3 S^1 \times S^3 . Specifically, we show that for any smoothly embedded 3 3 -manifold Y Y representing a generator of H 3 ( X ) H_3(X) , a suitable version of the Heegaard Floer d d invariant of Y Y , defined using twisted coefficients, is a diffeomorphism invariant of X X . We show how this invariant can be used to obstruct embeddings of certain types of 3 3 -manifolds, including those obtained as a connected sum of a rational homology 3 3 -sphere and any number of copies of S 1 × S 2 S^1 \times S^2 . We also give similar obstructions to embeddings in certain open 4 4 -manifolds, including exotic R 4 \mathbb {R}^4 s.

In this paper, we study the (in)sensitivity of the Khovanov functor to four-dimensional linking of surfaces. We prove that if $L$ and $L'$ are split links, and $C$ is a cobordism between $L$ and $L'$ that is the union of disjoint (but possibly linked) cobordisms between the components of $L$ and the components of $L'$, then the map on Khovanov homology induced by $C$ is completely determined by the maps induced by the individual components of $C$ and does not detect the linking between the components. As a corollary, we prove that a strongly homotopy-ribbon concordance (i.e., a concordance whose complement can be built with only 1- and 2-handles) induces an injection on Khovanov homology, which generalizes a result of the second author and Zemke. Additionally, we show that a non-split link cannot be ribbon concordant to a split link., Comment: 35 pages, 15 figures

We show that a ribbon concordance between two links induces an injective map on Khovanov homology., Published version; errors corrected from previous version. 5 pages, 1 figure

We study a class of 3-manifolds called strong L-spaces, which by definition admit a certain type of Heegaard diagram that is particularly simple from the perspective of Heegaard Floer homology. We provide evidence for the possibility that every strong L-space is the branched double cover of an alternating link in the three-sphere. For example, we establish this fact for a strong L-space admitting a strong Heegaard diagram of genus two via an explicit classification. We also show that there exist finitely many strong L-spaces with bounded order of first homology; for instance, through order eight, they are connected sums of lens spaces. The methods are topological and graph theoretic. We discuss many related results and questions., Comment: 33 pages, 9 figures

We use bordered Heegaard Floer homology to compute the tau invariant of a family of satellite knots obtained via twisted infection along two components of the Borromean rings, a generalization of Whitehead doubling. We show that tau of the resulting knot depends only on the two twisting parameters and the values of tau for the two companion knots. We also include some notes on bordered Heegaard Floer homology that may serve as a useful introduction to the subject., Comment: 72 pages, 30 figures, some in color. Version 2 (published version): made substantial revisions throughout, especially in Section 3, and added Appendix. Ancillary files include Mathematica notebooks for bordered Heegaard Floer homology computations

We introduce the notion of a strong L-space, a closed, oriented rational homology 3-sphere whose Heegaard Floer homology can be determined at the chain level. We prove that the fundamental group of a strong L-space is not left-orderable. Examples of strong L-spaces include the double branched covers of alternating links in S^3.

We use grid diagrams to give a combinatorial algorithm for computing the knot Floer homology of the pullback of a knot K in its m-fold cyclic branched cover Sigma^m(K), and we give computations when m=2 for over fifty three-bridge knots with up to eleven crossings., Comment: 30 pages, 2 figures, 1 long table. Added computations in Section 5; rewrote Section 6; corrected typos and minor mistakes

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 $\mathbb{Z}_2$., Published version. 46 pages, 5 figures

We investigate constraints on embeddings of a non-orientable surface in a $4$-manifold with the homology of $M \times I$, where $M$ is a rational homology $3$-sphere. The constraints take the form of inequalities involving the genus and normal Euler class of the surface, and either the Ozsv\'ath--Sazb\'o $d$-invariants or Atiyah--Singer $\rho$-invariants of $M$. One consequence is that the minimal genus of a smoothly embedded surface in $L(2p,q) \times I$ is the same as the minimal genus of a surface in $L(2p,q)$. We also consider embeddings of non-orientable surfaces in closed $4$-manifolds., Comment: Primary article by Adam Levine, Daniel Ruberman and Saso Strle, with an appendix by Ira Gessel. 54 pages, 6 figures

We show that the integer homology sphere obtained by splicing two nontrivial knot complements in integer homology sphere L-spaces has Heegaard Floer homology 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., 25 pages. Revised version, to appear in Crelle's Journal. Errors from the original version have been corrected

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., 58 pages, 18 figures. Published version, with updated references

We show that if K is any knot whose Ozsvath-Szabo concordance invariant tau(K) is positive, the all-positive Whitehead double of any iterated Bing double of K is topologically but not smoothly slice. We also show that the all-positive Whitehead double of any iterated Bing double of the Hopf link (e.g., the all-positive Whitehead double of the Borromean rings) is not smoothly slice; it is not known whether these links are topologically slice., 16 pages, 10 figures. v2: This is a substantial revision of v1. We eliminated Section 4 of v1 because it is superceded by arXiv:1008.3349. v3: corrected references

We use the Heegaard Floer obstructions defined by Grigsby, Ruberman, and Strle to show that forty-six of the sixty-seven knots through eleven crossings whose concordance orders were previously unknown have infinite concordance order., 5 pages, 3 tables