Adam Simon Levine
Department of Mathematics
Duke University
211 Physics Building, 120 Science Drive
Durham, NC 27708
Email: alevine at math dot duke dot edu
I am an associate professor at Duke University. My research is in lowdimensional topology, with a focus on Heegaard Floer homology, Khovanov homology, and their applications to knot theory, concordance, exotic 4manifolds, 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 (DMS2203860).
Papers

New constructions and invariants of closed exotic 4manifolds (with Tye Lidman and Lisa Piccirillo)
[abstract]
[pdf] Preprint.
In this article, we give new means of constructing and distinguishing closed exotic fourmanifolds. Using Heegaard Floer homology, we define new closed fourmanifold invariants that are distinct from the SeibergWitten and BauerFuruta 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 squarezero spheres.

A note on rationally slice knots
[abstract]
[pdf] New York Journal of Mathematics, to appear.
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 4manifold, 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 fourdimensional 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 homotopyribbon concordance (i.e., a concordance whose complement can be built with only 1 and 2handles) induces an injection on Khovanov homology, which generalizes a result of myself and Zemke. Additionally, we show that a nonsplit link cannot be ribbon concordant to a split link.

Khovanov homology detects the figureeight knot (with John A. Baldwin, Nathan Dowlin, Tye Lidman, and Radmila Sazdanovic)
[abstract]
[pdf] Bulletin of the London Mathematical Society 53 (2021), no. 3, 871876.

Khovanov homology and ribbon concordance (with Ian Zemke)
[abstract]
[pdf] Bulletin of the London Mathematical Society 51 (2019), no. 6, 10991103.
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, to appear.
Let \(K\) be a rationally nullhomologous knot in a 3manifold \(Y\), 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.

Simplyconnected, spineless 4manifolds (with Tye Lidman)
[abstract]
[pdf] Forum of Mathematics, Sigma,
7 (2019), e14.
We construct infinitely many smooth 4manifolds which are homotopy equivalent to \(S^2\) but do not admit a spine, i.e., a piecewiselinear embedding of \(S^2\) which realizes the homotopy equivalence. This is the remaining case in the existence problem for codimension2 spines in simplyconnected 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, 30893131.
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 nonlocallyflat piecewiselinear 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, 304981.
Using Heegaard Floer homology, we construct a numerical invariant for any smooth, oriented 4manifold \(X\) with the homology of \(S^1 \times S^3\). Specifically, we show that for any smoothly embedded 3manifold \(Y\) 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 3manifolds, including those obtained as a connected sum of a rational homology 3sphere and any number of copies of \(S^1 \times S^2\). We also give similar obstructions to embeddings in certain open 4manifolds, 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 wellknown 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 \(2^{l1}\) 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 Lspaces (with Joshua Greene)
[abstract]
[pdf]
Algebraic & Geometric Topology 16 (2016), 31673208.
We study a class of 3manifolds called strong Lspaces, 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 Lspace is the branched double cover of an alternating link in the threesphere. For example, we establish this fact for a strong Lspace admitting a strong Heegaard diagram of genus two via an explicit classification. We also show that there exist finitely many strong Lspaces 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.

Nonsurjective satellite operators and piecewiselinear 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 3sphere, the satellite knot with pattern \(P\) and companion \(K\) is not smoothly slice in any homology 4ball. As a consequence, we obtain a knot in a homology 3sphere that does not bound a piecewiselinear disk in any contractible 4manifold.

Generalized Heegaard Floer correction terms (with Daniel Ruberman)
[abstract]
[pdf]
Proceedings of Gökova GeometryTopology Conference 2013, 7696.
We make use of the action of \(H_1(Y)\) in Heegaard Floer homology to generalize the OzsváthSazbó correction terms for 3manifolds 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.

Nonorientable 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, 439494.
We investigate constraints on embeddings of a nonorientable surface in a 4manifold with the homology of \(M \times I\), where \(M\) is a rational homology 3sphere. The constraints take the form of inequalities involving the genus and normal Euler class of the surface, and either the OzsváthSazbó \(d\)invariants or AtiyahSinger \(\rho\)invariants of \(M\). One consequence is that the minimal genus of a smoothly embedded, nonorientable surface in \(L(2k,q) \times I\) is the same as the minimal genus of an embedded, nonorientable surface in \(L(2k,q)\), which was computed by Bredon and Wood. We also consider embeddings of nonorientable surfaces in closed 4manifolds.

Splicing knot complements and bordered Floer homology (with Matthew Hedden)
[abstract]
[pdf]
Journal für die reine und angewandte Mathematik 720 (2016), 129154.
We show that the integer homology sphere obtained by splicing two nontrivial knot complements in integer homology sphere Lspaces has Heegaard Floer homology rank strictly greater than one. In particular, splicing the complements of nontrivial knots in the 3sphere never produces an Lspace. The proof uses bordered Floer homology.

Strong Lspaces and leftorderability (with Sam Lewallen)
[abstract]
[pdf]
Mathematical Research Letters 19 (2012), no. 6, 12371244.
A strong Lspace 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 Lspaces
include double branched covers of alternating links in \(S^3\). We show using an
elementary argument that the fundamental group of any strong Lspace is not
leftorderable.

A combinatorial spanning tree model for knot Floer homology (with John Baldwin)
[abstract]
[pdf]
Advances in Mathematics 231 (2012), 18861939.
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 deltagraded 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 BingWhitehead doubles
[abstract]
[pdf]
Journal of Topology 5 (2012), 713726.
We show that if \(K\) is any knot whose OzsváthSzabó concordance invariant \(\tau(K)\) is positive, the allpositive Whitehead double of any iterated Bing double of \(K\) is topologically but not smoothly slice. We also show that the allpositive Whitehead double of any iterated Bing double of the Hopf link (e.g., the allpositive 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), 651712.
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 fortysix of the sixtyseven 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), 11631190.
We use grid diagrams to give a combinatorial algorithm for computing the knot Floer homology of the pullback of a knot \(K\) in its mfold cyclic branched cover \(\Sigma^m(K)\), and we give computations when \(m=2\) for over fifty threebridge knots with up to eleven crossings.
Teaching
In Fall 2023, I am teaching a minicourse, Introduction to 4Manifolds, (Tuesdays and Thursdays, 1:252:40pm, November 2  December 5).
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)

Math 611: Algebraic Topology I (Fall 2018)

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

Math 620: Smooth Manifolds (Fall 2020)

Math 69010: Topics in Topology (Fall 2017)

Math 790: Knot Homologies (minicourse) (Spring 2021).
Talks
Here are videos and/or slides from a few of my talks:

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

"Using Heegaard Floer homology to construct interesting 4manifolds"
(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:
Part I
Part II
Part III
Part IV
Winter School on Homology Theories in LowDimensional Topology, Isaac Newton Institute, Cambridge, England, January 1619, 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 piecewiselinear concordance"
(video)
(related slides)
Synchronizing Smooth and Topological 4Manifolds, Banff International Research Station, February 26, 2016

"Nonorientable surfaces in 3 and 4manifolds" (slides)
University of Virginia Mathematics Colloquium, October 31, 2013

"Lspaces, taut foliations, leftorderability, 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
Miscellaneous