Publications and Preprints:
On knot Floer homology and cabling Algebraic & Geometric Topology 5 (2005), 1197–1222. Published version or Arxiv.
Knot Floer homology of Whitehead doubles Geometry & Topology 11 (2007), 2277–2338. Published version or Arxiv.
The Ozsváth-Szabó and Rasmussen concordance invariants are not equal Joint with Philip Ording.
American Journal of Mathematics, 130 (2008), no. 2, 441–453.
Published version or Arxiv.
Grid Diagrams for Lens Spaces and Combinatorial Knot Floer Homology Joint with Kenneth L. Baker J. Elisenda Grigsby.
International Mathematics Research Notices, 2008, no. 10, Art. ID rnm024, 39 pp.
Published version or Arxiv.
An Ozsváth–Szabó Floer homology invariant of knots in a contact manifold
Advances in Mathematics, 219 (2008), no. 1, 89–117.
Published version or Arxiv.
Some remarks on cabling, contact structures, and complex curves
Proceedings of Gökova Geometry-Topology Conference 2007, 49–59.
Published version or Arxiv.
On knot Floer homology and cabling. II.
International Mathematics Research Notices, 2009, no. 12, 2248–2274.
Published version or Arxiv.
Khovanov homology of the 2-cable detects the unknot.
Mathematical Research Letters
Published version or Arxiv.
Notions of positivity and the Ozsváth-Szabó concordance invariant.
Journal of Knot Theory and its Ramifications, 19 (2010), no. 5, 617–629.
Published version or Arxiv.
Manifolds with small Heegaard Floer ranks.
Joint with Yi Ni.
Geometry & Topology, 14 (2010), no. 3, 1479–1501.
Published version or Arxiv.
Does Khovanov homology detect the unknot?
Joint with Liam Watson.
American Journal of Mathematics, 132 (2010), no. 5
Published version or Arxiv.
On Floer homology and the Berge conjecture on knots admitting lens space surgeries.
Transactions of the American Mathematical Society, 363 (2011), no. 2, 949–968.
Published version or Arxiv.
Chern-Simons invariants, SO(3) instantons, and ℤ/2 homology cobordism.
Joint with Paul Kirk
Chern-Simons gauge theory: 20 years after, 83–114, AMS/IP Studies in Advanced Mathematics 50, Amer. Math. Soc., Providence, RI, 2011.
Published version or Arxiv.
Non-slice linear combinations of algebraic knots.
Joint with Paul Kirk and Charles Livingston.
Journal of the European Mathematical Society, 14 (2012), no. 4, 1181–1208.
Published version or Arxiv.
Topologically slice knots with nontrivial Alexander polynomial.
Joint with Charles Livingston and Daniel Ruberman
Advances in Mathematics, 231 (2012), no. 2, 913–939.
Published version or Arxiv.
Instantons, concordance, and Whitehead doubling.
Joint with Paul Kirk
Journal of Differential Geometry, 91 (2012), no. 2, 281–319.
Published version or Arxiv.
Knot concordance and homology cobordism.
Joint with Tim Cochran, Bridget Franklin, and Peter Horn.
Proceedings of the American Mathematical Society, 141 (2013), no. 6, 2193–2208.
Published version or Arxiv.
Dehn surgery, rational open books and knot Floer homology.
Joint with Olga Plamenevskaya.
Algebraic & Geometric Topology, 13 (2013) 1815–1856.
Published version or Arxiv.
On sutured Floer homology and the equivalence of Seifert surfaces.
Joint with Andras Juhasz and Sucharit Sarkar.
Algebraic & Geometric Topology, 13 (2013) 505–548.
Published version or Arxiv.
Khovanov module and the detection of unlinks.
Joint with Yi Ni.
Geometry & Topology, 17 (2013) 3027–3076.
Published version or Arxiv.
The pillowcase and perturbations of traceless representations of knot groups.
Joint with Christopher Herald and Paul Kirk.
Geometry & Topology, 18 (2014) 211–287.
Published version or Arxiv.
Topologically slice knots of smooth concordance order two.
Joint with Se-Goo Kim and Charles Livingston.
Journal of Differential Geometry, 102 (2016), no. 3, 353–393.
Published version or Arxiv.
Splicing knot complements and bordered Floer homology.
Joint with Adam Levine.
Journal für die reine und angewandte Mathematik, 720 (2016), 129–154.
Published version or Arxiv.
Plane algebraic curves of arbitrary genus via Heegaard Floer homology.
Joint with Maciej Borodzik and Charles Livingston.
Commentarii Mathematici Helvetici, 92 (2017), no. 2, 215–256.
Published version or Arxiv.
Floer homology and fractional Dehn twists.
Joint with Thomas Mark.
Advances in Mathematics, 324 (2018), 1–39.
Published version or Arxiv.
On the geography and botany of knot Floer homology.
Joint with Liam Watson.
Selecta Mathematica. New Series, 24 (2018), no. 2, 997–1037.
Published version or Arxiv.
The Υ function of L-space knots is a Legendre transform.
Joint with Maciej Borodzik.
Mathematical Proceedings of the Cambridge Philosophical Society, 164 (2018), no. 3, 401–411.
Published version or Arxiv.
The pillowcase and traceless representations of knot groups II: a Lagrangian-Floer theory in the pillowcase.
Joint with Christopher Herald and Paul Kirk.
Journal of Symplectic Geometry 16 (2018) 3, 721-815.
Published version or Arxiv.
On the functoriality of Khovanov-Floer theories.
Joint with John Baldwin and Andrew Lobb.
On the functoriality of Khovanov-Floer theories. Advances in Mathematics 345 (2019), 1162–1205.
Published version or Arxiv.
Irreducible 3-manifolds that cannot be obtained by 0-surgery on a knot.
Joint with Min Hoon Kim, Thomas Mark, and Kyungbae Park.
Transactions of the American Mathematical Society. 372(11) (2019), 7619--7638.
Published version or Arxiv.
The Fukaya category of the pillowcase, traceless character varieties, and Khovanov Cohomology.
Joint with Christopher Herald, Matthew Hogancamp, and Paul Kirk.
Transactions of the American Mathematical Society. 363(2) (2011) 949--968.
Published version or
Arxiv.
Satellites of Infinite Rank in the Smooth Concordance Group.
Joint with Juanita Pinzón-Caicedo.
Inventiones Mathematicae. DOI 10.1007/s00222-020-01026-w
Published version or
Arxiv.
4-dimensional aspects of tight contact 3-manifolds
Joint with Katherine Raoux.
Proceedings of the National Academy of Sciences. vol. 118 no. 22 e2025436118
Published version or
Arxiv.
Corks, involutions, and Heegaard Floer homology
Joint with Irving Dai and Abhishek Mallick.
Journal of the European Mathematical Society, to appear.
Arxiv.
Surgery formulae for knot Floer homology
Joint with Adam Levine.
Submitted.
Arxiv.
Knot Floer homology and relative adjunction inequalities
Joint with Katherine Raoux.
Submitted.
Arxiv.
On naturality of the Ozsvath-Szabo contact invariant
Joint with Lev Tovstopyat-Nelip.
Submitted.
Arxiv.
In Preparation:
A link concordance group from knots in connected sums of S1xS2
Joint with Miriam Kuzbary.
Abstract: We define a group structure on link concordance classes and study its properties.
Knot theory and algebraic curves in subcritical Stein domains
Abstract: I characterize knots that bound algebraic curves in subcritical Stein domains as closures of quasipositive braids in the braid group of a multi-punctured sphere.
Quasipositive braids in open books as boundaries of complex curves.
Joint with I. Baykur, J. Etnyre, K.Hayden, K. Kawamuro, and J. Van Horn-Morris.
Abstract: We show that closures of general quasipositive braids for a Stein fillable open book bound complex curves in a Stein filling, completing the topological scharacterization of knots in arbitrary 3-manifolds which bound complex curves in a Stein filling.
Murasugi sum and the top group of knot Floer homology.
Joint with Sucharit Sarkar and Zhechi Cheng.
Abstract: Work of Ni showed that the rank of the top group of knot Floer homology is multiplicative under Murasugi sums. We generalize this to show that the top group of a Murasugi sum is the graded tensor product of the top groups of the summands.
Research Grants and Fellowships:
National Science Foundation Research Grant DMS-1709016
Floer Homology, Concordance, and Complex Curves
2017-2020
Alfred P. Sloan Research Fellowship
2011-2013
National Science Foundation CAREER Grant DMS-1150872
Floer Homology and Low-Dimensional Topology
2012-2018
National Science Foundation Research Grant DMS-0906258
Knots and surfaces in three- and four-manifolds: Applications of symplectic topology and quantum algebra to low dimensional topology
2009-2012
National Science Foundation Research Grant DMS-0706979
Low dimensional topology and invariants from symplectic geometry, gauge theory, and quantum algebra
2007-2009
National Science Foundation Postdoctoral Research Fellowship DMS-0503335
2005-2007