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 assistant professor at Duke University. My research is in low-dimensional topology, with a focus on Heegaard Floer homology and its applications to knot theory, concordance, surfaces in 3- and 4-manifolds, and other areas. My CV can be found here.
I am partially supported by an NSF Topology grant (DMS-1707795).
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] Preprint.
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, to appear.
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 \(\mathbb{Z}/2\mathbb{Z}\).
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, 1237-1234.
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.
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.
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.
Teaching
In Spring 2019, I am teaching Math 411: Topology (TTh 8:30-9:45, Gross 318).