Journal articleQuantum Topology · January 1, 2024
Let K be a rationally null-homologous knot in a 3-manifold Y, equipped with a non-zero framing λ, and let Yλ(K) denote the result of λ-framed surgery on Y. Ozsváth and Szabó gave a formula for the Heegaard Floer homology groups of Yλ ...
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Preprint · July 16, 2023
In this article, we give new means of constructing and distinguishing closed exotic four-manifolds. Using Heegaard Floer homology, we define new closed four-manifold invariants that are distinct from the Seiberg--Witten and Bauer--Furuta invariants and can ...
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Journal articleNew York Journal of Mathematics · January 1, 2023
Kawauchi proved that every strongly negative amphichiral knot (Formula Presented) bounds a smoothly embedded disk in some rational homology ball VK, whose construction a priori depends on K. We show that VK is inde-pendent of K up to ...
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Preprint · December 25, 2022
Kawauchi proved that every strongly negative amphichiral knot $K \subset S^3$ bounds a smoothly embedded disk in some rational homology ball $V_K$, whose construction a priori depends on $K$. We show that $V_K$ is independent of $K$ up to diffeomorphism. T ...
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Journal articleDuke Mathematical Journal · October 15, 2022
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Let CbZ 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 sm ...
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Journal articleJournal of Topology · September 1, 2022
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In this paper, we study the (in)sensitivity of the Khovanov functor to 4-dimensional linking of surfaces. We prove that if (Formula presented.) and (Formula presented.) are split links, and (Formula presented.) is a cobordism between (Formula presented.) a ...
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Journal articleBulletin of the London Mathematical Society · June 1, 2021
We use Dowlin's spectral sequence from Khovanov homology to knot Floer homology to prove that reduced Khovanov homology with rational coefficients detects the figure-eight knot. ...
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Preprint · September 7, 2020
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) cobo ...
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Preprint · July 2, 2020
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. ...
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Journal articleTransactions of the American Mathematical Society · January 1, 2020
We prove that twisted correction terms in Heegaard Floer homology provide lower bounds on the Thurston norm of certain cohomology classes determined by the strong concordance class of a 2-component link L in S3. We then specialise this procedure ...
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Journal articleBulletin of the London Mathematical Society · December 1, 2019
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We show that a ribbon concordance between two links induces an injective map on Khovanov homology. ...
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Preprint · January 8, 2019
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 ...
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Journal articleForum of Mathematics Sigma · January 1, 2019
We construct infinitely many compact, smooth 4-manifolds which are homotopy equivalent to S2 but do not admit a spine (that is, a piecewise linear embedding of S2 that realizes the homotopy equivalence). This is the remaining case in ...
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Preprint · March 5, 2018
We construct infinitely many smooth 4-manifolds which are homotopy equivalent to $S^2$ but do not admit a spine, i.e., a piecewise-linear embedding of $S^2$ which realizes the homotopy equivalence. This is the remaining case in the existence problem for co ...
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Preprint · January 23, 2018
Let $\widehat{\mathcal{C}}_{\mathbb{Z}}$ 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 ...
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Journal articleJournal of Knot Theory and Its Ramifications · February 1, 2017
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A well-known conjecture states that for any l-component link L in S3, the rank of the knot Floer homology of L (over any field) is less than or equal to 2l-1 times the rank of the reduced Khovanov homology of L. In this paper, we describe a framework that ...
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Journal articleAlgebraic and Geometric Topology · December 15, 2016
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–sp ...
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Journal articleJournal Fur Die Reine Und Angewandte Mathematik · November 1, 2016
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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 ...
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Preprint · October 11, 2016
Using Heegaard Floer homology, we construct a numerical invariant for any smooth, oriented $4$-manifold $X$ with the homology of $S^1 \times S^3$. Specifically, we show that for any smoothly embedded $3$-manifold $Y$ representing a generator of $H_3(X)$, a ...
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Journal articleForum of Mathematics Sigma · January 1, 2016
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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 ...
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Preprint · December 16, 2015
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 ...
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Journal articleGeometry and Topology · February 27, 2015
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We investigate constraints on embeddings of a nonorientable surface in a 4–manifold with the homology of M × 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 ...
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Preprint · November 23, 2014
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-sp ...
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Preprint · May 5, 2014
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 obtai ...
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Preprint · March 10, 2014
We make use of the action of $H_1(Y)$ in Heegaard Floer homology to generalize the Ozsváth-Szabó correction terms for $3$-manifolds with standard $\operatorname{HF}^\infty$. We establish the basic properties of these invariants: conjugation invariance, beh ...
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Preprint · October 31, 2013
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 ...
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Preprint · October 26, 2012
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 ...
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Journal articleAdvances in Mathematics · October 1, 2012
We iterate Manolescu's unoriented skein exact triangle in knot Floer homology with coefficients in the field of rational functions over Z/2Z. The result is a spectral sequence which converges to a stabilized version of δ-graded knot Floer homology. The (E ...
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Journal articleJournal of Topology · January 1, 2012
We show that if K is any knot whose Ozsváth-Szabó concordance invariant T (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 an ...
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Journal articleJournal of Topology · January 1, 2012
We use bordered Heegaard Floer homology to compute the τ 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 τ of the resulting knot depen ...
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Journal articleMathematical Research Letters · January 1, 2012
We introduce the notion of a strong L-space, a closed, oriented threemanifold admitting a Heegaard diagram whose associated Heegaard Floer complex has rank equal to the order of the first homology of the manifold. Examples of strong Lspaces include the bra ...
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Preprint · October 3, 2011
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 ...
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Preprint · May 26, 2011
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 ...
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Preprint · August 19, 2010
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 d ...
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Preprint · December 28, 2009
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 a ...
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Journal articleAlgebraic and Geometric Topology · December 1, 2008
We use grid diagrams to give a combinatorial algorithm for computing the knot Floer homology of the pullback of a knot K ⊂ S3 in its m-fold cyclic branched cover Σm(K), and we give computations when m=2 for over fifty three - bridge k ...
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Preprint · May 15, 2008
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. ...
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Preprint · September 10, 2007
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 c ...
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https://orcid.org/0000-0002-9084-5124