Adam Simon Levine

Brandeis University
Mathematics Department, Office 96-205
415 South Street
Waltham, MA 02453
E-mail: levinea at brandeis.edu
Telephone: (781) 736-3056

I am an NSF Mathematical Sciences Postdoctoral Research Fellow at Brandeis University. I study low-dimensional topology, specifically Heegaard Floer homology and its applications to concordance and other areas of knot theory. My curriculum vitae can be found here.

Papers

Strong L-spaces and left-orderability (with Sam Lewallen)
[abstract] [pdf]
A strong L-space is a rational homology sphere Y that admits a Heegaard diagram whose associated Heegaard Floer complex has exactly |H^2(Y;Z)| 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]
We iterate Manolescu's unoriented skein exact triangle in knot Floer homology with coefficients in the fraction field of the group ring (Z/2Z)[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, to appear.
We show that if K is any knot whose Ozsváth-Szabó concordance invariant τ(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, to appear.
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. We show that τ 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, to appear.
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 Σ^m(K), and we give computations when m=2 for over fifty three-bridge knots with up to eleven crossings.

Thesis

My Ph.D. thesis is Applications of Heegaard Floer homology to knot and link concordance, supervised by Peter Ozsváth at Columbia University. It consists in large part of my first four papers, listed above. Here is a summary in sonnet form:

Our goal is one whose application's nice
For smooth four-manifold topology:
To tell if certain knots and links are slice
With bordered Heegaard Floer homology.

We seek concordance data that detect
Some links obtained by Whitehead doublings,
As well as knots we get when we infect
Along two of the three Borromean rings.

Some lengthy work with bordered Floer then proves
How τ for satellites like these is found.
We see, by this result and cov'ring moves,
That smooth slice disks our links can never bound.

The theorem's proved, the dissertation's done,
But all the work ahead has just begun.

Teaching