Adam Simon Levine

Department of Mathematics
Duke University
211 Physics Building, 120 Science Drive
Durham, NC 27708
E-mail: alevine at math dot duke dot edu

I am an associate professor at Duke University. My research is in low-dimensional topology, with a focus on Heegaard Floer homology, Khovanov homology, and their applications to knot theory, concordance, exotic 4-manifolds, and other areas. I am also the Director of Undergraduate Studies for the mathematics department. My CV can be found here.

I am partially supported by an NSF Topology grant (DMS-2203860).

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Papers

• New constructions and invariants of closed exotic 4-manifolds (with Tye Lidman and Lisa Piccirillo)
  [abstract] [pdf] Preprint.

  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 remain distinct in covers. Our constructions include exotic definite manifolds with fundamental group \(\mathbb Z/2\), infinite families of exotic manifolds that are related by knot surgeries on Alexander polynomial 1 knots, and exotic manifolds that contain square-zero spheres.

• A note on rationally slice knots
  [abstract] [pdf] New York Journal of Mathematics 29 (2023), 1363–72.

  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. Thus, a single 4-manifold, along with connected sums thereof, accounts for all known examples of knots that are rationally slice but not slice.

• Khovanov homology and cobordisms between split links (with Onkar Singh Gujral)
  [abstract] [pdf] Journal of Topology 15 (2022), no. 3, 973–1016.

  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 myself and Zemke. Additionally, we show that a non-split link cannot be ribbon concordant to a split link.

• Khovanov homology detects the figure-eight knot (with John A. Baldwin, Nathan Dowlin, Tye Lidman, and Radmila Sazdanovic)
  [abstract] [pdf] Bulletin of the London Mathematical Society 53 (2021), no. 3, 871-876.

  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.

• Khovanov homology and ribbon concordance (with Ian Zemke)
  [abstract] [pdf] Bulletin of the London Mathematical Society 51 (2019), no. 6, 1099-1103.

  We show that a ribbon concordance between two links induces an injective map on Khovanov homology.

• A surgery formula for knot Floer homology (with Matthew Hedden)
  [abstract] [pdf] Quantum Topology 15 (2024), 229-336.

  Let \(K\) be a rationally null-homologous knot in a 3-manifold \(Y\), equipped with a nonzero framing \(\lambda\), and let \(Y_\lambda(K)\) denote the result of \(\lambda\)-framed surgery on \(Y\). Ozsváth and Sazbó gave a formula for the Heegaard Floer homology groups of \(Y_\lambda(K)\) 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 \(K_\lambda\) in \(Y_\lambda\), i.e., the core circle of the surgery solid torus.

• Simply-connected, spineless 4-manifolds (with Tye Lidman)
  [abstract] [pdf] Forum of Mathematics, Sigma, 7 (2019), e14.

  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 codimension-2 spines in simply-connected manifolds. The obstruction comes from the Heegaard Floer \(d\) invariants.

• Knot concordance in homology cobordisms (with Jennifer Hom and Tye Lidman)
  [abstract] [pdf] Duke Mathematical Journal 171 (2022), no. 15, 3089-3131.

  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 from the smooth knot concordance group \(\mathcal{C}\) to \(\widehat{\mathcal{C}}_{\mathbb{Z}}\) 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.

• Heegaard Floer invariants in codimension one (with Daniel Ruberman)
  [abstract] [pdf] Transactions of the AMS 371 (2019), no. 5, 3049-81.

  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 suitable version of the Heegaard Floer d invariant of \(Y\), defined using twisted coefficients, is a diffeomorphism invariant of \(X\). We show how this invariant can be used to obstruct embeddings of certain types of 3-manifolds, including those obtained as a connected sum of a rational homology 3-sphere and any number of copies of \(S^1 \times S^2\). We also give similar obstructions to embeddings in certain open 4-manifolds, including exotic \(\mathbb{R}^4\)s.

• Khovanov homology and knot Floer homology for pointed links (with John Baldwin and Sucharit Sarkar)
  [abstract] [pdf] Tim Cochran Memorial Volume, Journal of Knot Theory and its Ramifications 26 (2017), 1740004.

  A well-known conjecture states that for any \(l\)-component link \(L \subset S^3\), 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 \(\mathbb{Z}/2\mathbb{Z}\).

• Strong Heegaard diagrams and strong L-spaces (with Joshua Greene)
  [abstract] [pdf] Algebraic & Geometric Topology 16 (2016), 3167-3208.

  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.

• Non-surjective satellite operators and piecewise-linear concordance
  [abstract] [pdf] Forum of Mathematics, Sigma 4 (2016).

  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 contractible 4-manifold.

• Generalized Heegaard Floer correction terms (with Daniel Ruberman)
  [abstract] [pdf] Proceedings of Gökova Geometry-Topology Conference 2013, 76-96.

  We make use of the action of \(H_1(Y)\) in Heegaard Floer homology to generalize the Ozsváth-Sazbó correction terms for 3-manifolds with standard \(HF^\infty\). We establish the basic properties of these invariants: conjugation invariance, behavior under orientation reversal, additivity, and spin^c rational homology cobordism invariance.

• Non-orientable surfaces in homology cobordisms (with Daniel Ruberman and Sašo Strle, and with an appendix by Ira M. Gessel)
  [abstract] [pdf] Geometry & Topology 19 (2015), no. 1, 439-494.

  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 Ozsváth-Sazbó \(d\)-invariants or Atiyah-Singer \(\rho\)-invariants of \(M\). One consequence is that the minimal genus of a smoothly embedded, non-orientable surface in \(L(2k,q) \times I\) is the same as the minimal genus of an embedded, non-orientable surface in \(L(2k,q)\), which was computed by Bredon and Wood. We also consider embeddings of non-orientable surfaces in closed 4-manifolds.

• Splicing knot complements and bordered Floer homology (with Matthew Hedden)
  [abstract] [pdf] Journal für die reine und angewandte Mathematik 720 (2016), 129-154.

  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.

• Strong L-spaces and left-orderability (with Sam Lewallen)
  [abstract] [pdf] Mathematical Research Letters 19 (2012), no. 6, 1237-1244.

  A strong L-space is a rational homology sphere Y that admits a Heegaard diagram whose associated Heegaard Floer complex has exactly \(\lvert H^2(Y;\mathbb{Z})\rvert\) generators, a rather rigid combinatorial condition. Examples of strong L-spaces include double branched covers of alternating links in \(S^3\). We show using an elementary argument that the fundamental group of any strong L-space is not left-orderable.

• A combinatorial spanning tree model for knot Floer homology (with John Baldwin)
  [abstract] [pdf] Advances in Mathematics 231 (2012), 1886-1939.

  We iterate Manolescu's unoriented skein exact triangle in knot Floer homology with coefficients in the fraction field of the group ring \((\mathbb{Z}/2\mathbb{Z})[\mathbb{Z}]\). 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.

• Slicing mixed Bing-Whitehead doubles
  [abstract] [pdf] Journal of Topology 5 (2012), 713-726.

  We show that if \(K\) is any knot whose Ozsváth-Szabó 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.

• Knot doubling operators and bordered Heegaard Floer homology
  [abstract] [pdf] Journal of Topology 5 (2012), 651-712.

  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. 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.

• On knots with infinite concordance order
  [abstract] [pdf] Journal of Knot Theory and its Ramifications 21 (2012), no. 12, 1250014.

  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.

• Computing knot Floer homology in cyclic branched covers
  [abstract] [pdf] Algebraic & Geometric Topology 8 (2008), 1163--1190.

  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.

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Teaching

In Fall 2024, I am teaching a minicourse, Link Homology Theories and Immersed Curves, (Mondays and Wednesdays, 10:05-11:20am, October 30 - December 4).

I have previously taught the following courses at Duke:

• Math 221: Linear Algebra (Fall 2017, Fall 2019)

• Math 222: Vector Calculus (Spring 2023)

• Math 401: Abstract Algebra (Fall 2021)

• Math 411: Topology (Spring 2019, Spring 2022, Spring 2024)

• Math 611: Algebraic Topology I (Fall 2018)

• Math 612: Algebraic Topology II (Spring 2020, Spring 2021)

• Math 620: Smooth Manifolds (Fall 2020)

• Math 690-10: Topics in Topology (Fall 2017)

• Math 790: Knot Homologies (minicourse) (Spring 2021).

• Math 790: Introduction to 4-Manifolds (minicourse) (Fall 2023).

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Talks

Here are videos and/or slides from a few of my talks:

• "Heegaard Floer homology and closed exotic 4-manifolds" (video)
  Duke Geometry/Topology Seminar, September 11, 2023

• "Using Heegaard Floer homology to construct interesting 4-manifolds" (video)
  Using Quantum Invariants to Do Interesting Topology, Casa Matematica Oaxaca, October 24, 2022

• "Ribbon concordance and link homology theories" (video) (slides)
  Categorification Learning Seminar, June 4, 2020

• Lecture series on Heegaard Floer homology
  Winter School on Homology Theories in Low-Dimensional Topology, Isaac Newton Institute, Cambridge, England, January 16-19, 2017

• "Khovanov homology and knot Floer homology for pointed links" (slides)
  AMS Southeastern Sectional Meeting, North Carolina State University, November 13, 2016

• "Satellite operators and piecewise-linear concordance" (video) (related slides)
  Synchronizing Smooth and Topological 4-Manifolds, Banff International Research Station, February 26, 2016

• "Non-orientable surfaces in 3- and 4-manifolds" (slides)
  University of Virginia Mathematics Colloquium, October 31, 2013

• "L-spaces, taut foliations, left-orderability, and incompressible tori" (slides)
  47th Annual Spring Topology and Dynamics Conference, Central Connecticut State University, March 24, 2013

• "Combinatorial spanning tree models for knot homologies" (slides)
  Knots in Washington XXXIII, George Washington University, December 2, 2011

• "Bordered Heegaard Floer homology and knot doubling operators" (slides)
  Knot Concordance and Homology Cobordism Workshop, Wesleyan University, July 21, 2010

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Miscellaneous

• Calendar of geometry/topology seminars in the Triangle area: Web ICal

• Video of my song parody "I Am the Very Model of a Heegaard Floer Homologist," which I performed at the Princeton Low-Dimensional Topology Workshop in June 2015:

• My Ph.D. thesis, plus a version in sonnet form.

• My entries in arXiv, MathSciNet, Mathematics Genealogy Project.

• My crossword puzzles page.

• Source sheets from the course on "Jewish Geometry" that I taught at the National Havurah Committee Summer Institute in 2018:
  Part 1: The Diameter and Circumference of a Circle
  Part 2: Square Roots and the Pythagorean Theorem
  Part 3: The Great Circle Route to Jerusalem