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Adam S. Levine

Associate Professor of Mathematics
Mathematics
120 Science Drive, Durham, NC 27708
Office hours Please email me for office hours.  

Selected Publications


A surgery formula for knot Floer homology

Journal Article Quantum 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 (K) in terms of the kn ... Full text Cite

A note on rationally slice knots

Journal Article New 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 diffeomorphism. Thus, ... Cite

A note on rationally slice knots

Preprint · December 25, 2022 Link to item Cite

KNOT CONCORDANCE IN HOMOLOGY COBORDISMS

Journal Article Duke Mathematical Journal · October 15, 2022 Featured Publication 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 smooth knot concordance ... Full text Cite

Khovanov homology and cobordisms between split links

Journal Article Journal of Topology · September 1, 2022 Featured Publication 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 ... Full text Cite

Khovanov homology detects the figure-eight knot

Journal Article Bulletin 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. ... Full text Cite

Heegaard floer homology and concordance bounds on the Thurston norm

Journal Article Transactions 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 to knots i ... Full text Cite

Khovanov homology and ribbon concordances

Journal Article Bulletin of the London Mathematical Society · December 1, 2019 Featured Publication We show that a ribbon concordance between two links induces an injective map on Khovanov homology. ... Full text Cite

SIMPLY CONNECTED, SPINELESS 4-MANIFOLDS

Journal Article Forum 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 the existence problem ... Full text Open Access Cite

Indivisible

Journal Article Mathematical Intelligencer · January 1, 2019 Full text Open Access Cite

Indivisible

Journal Article MATHEMATICAL INTELLIGENCER · 2019 Full text Cite

Heegaard Floer invariants in codimension one

Journal Article Transactions of the American Mathematical Society · August 9, 2017 Full text Cite

Khovanov homology and knot Floer homology for pointed links

Journal Article Journal of Knot Theory and its Ramifications · February 1, 2017 Featured Publication 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 ... Full text Open Access Cite

A surgery formula for knot Floer homology

Journal Article Quantum 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 (K) in terms of the kn ... Full text Cite

A note on rationally slice knots

Journal Article New 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 diffeomorphism. Thus, ... Cite

A note on rationally slice knots

Preprint · December 25, 2022 Link to item Cite

KNOT CONCORDANCE IN HOMOLOGY COBORDISMS

Journal Article Duke Mathematical Journal · October 15, 2022 Featured Publication 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 smooth knot concordance ... Full text Cite

Khovanov homology and cobordisms between split links

Journal Article Journal of Topology · September 1, 2022 Featured Publication 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 ... Full text Cite

Khovanov homology detects the figure-eight knot

Journal Article Bulletin 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. ... Full text Cite

Heegaard floer homology and concordance bounds on the Thurston norm

Journal Article Transactions 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 to knots i ... Full text Cite

Khovanov homology and ribbon concordances

Journal Article Bulletin of the London Mathematical Society · December 1, 2019 Featured Publication We show that a ribbon concordance between two links induces an injective map on Khovanov homology. ... Full text Cite

SIMPLY CONNECTED, SPINELESS 4-MANIFOLDS

Journal Article Forum 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 the existence problem ... Full text Open Access Cite

Indivisible

Journal Article Mathematical Intelligencer · January 1, 2019 Full text Open Access Cite

Indivisible

Journal Article MATHEMATICAL INTELLIGENCER · 2019 Full text Cite

Heegaard Floer invariants in codimension one

Journal Article Transactions of the American Mathematical Society · August 9, 2017 Full text Cite

Khovanov homology and knot Floer homology for pointed links

Journal Article Journal of Knot Theory and its Ramifications · February 1, 2017 Featured Publication 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 ... Full text Open Access Cite

Strong Heegaard diagrams and strong L–spaces

Journal Article Algebraic 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 ... Full text Open Access Cite

Splicing knot complements and bordered floer homology

Journal Article Journal fur die Reine und Angewandte Mathematik · November 1, 2016 Featured Publication 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 ... Full text Open Access Cite

NONSURJECTIVE SATELLITE OPERATORS and PIECEWISE-LINEAR CONCORDANCE

Journal Article Forum of Mathematics, Sigma · January 1, 2016 Featured Publication 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 ... Full text Open Access Cite

Nonorientable surfaces in homology cobordisms

Journal Article Geometry and Topology · February 27, 2015 Featured Publication 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 ... Full text Open Access Cite

Generalized Heegaard Floer correction terms

Journal Article Proceedings of the Gökova Geometry-Topology Conference 2013 · May 12, 2014 Open Access Link to item Cite

A combinatorial spanning tree model for knot Floer homology

Journal Article Advances 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 ... Full text Open Access Cite

On knots with infinite smooth concordance order

Journal Article Journal of Knot Theory and its Ramifications · February 1, 2012 We use the Heegaard Floer obstructions defined by Grigsby, Ruberman, and Strle to show that forty-five of the sixty-six knots through eleven crossings whose concordance orders were previously unknown have infinite concordance order. © 2012 World Scientific ... Full text Cite

Slicing mixed bing-whitehead doubles

Journal Article Journal 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 ... Full text Cite

Knot doubling operators and bordered heegaard floer homology

Journal Article Journal 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 ... Full text Cite

Strong L-spaces and left-orderability

Journal Article Mathematical 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 ... Full text Cite

Computing knot Floer homology in cyclic branched covers

Journal Article Algebraic 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 knots with up to eleven ... Full text Cite