Research

 
 

Papers

  1. Heegaard Floer homology of (n,n)-torus links: computations and questions

     We partially compute the Heegaard Floer link invariant of (n,n) torus links and give a conjecture for the complete invariant.  This paper has several open questions; please let me know if you answer them!

      (arXiv1208:0394)

  1. Legendrian Contact Homology in Seifert Fibered Spaces

     We define a contact homology-type invariant for Legendrian knots in Seifert fibered spaces equipped with transverse, S1-invariant contact structures.

      (To appear in Quantum Topology, joint with J. Sabloff)

  1. Rational Seifert Surfaces in Seifert Fibered Spaces 

     We construct rational Seifert surfaces for knots in Seifert fibered spaces.  As an application, we give combinatorial formulae for computing the rational Thurston-Bennequin and rotation numbers of rationally null-homologous knots in contact Seifert fibered spaces.

     Pacific Journal of Mathematics 258-1 (2012), 1990221, joint with J. Sabloff.

  1. Legendrian grid number one knots and augmentations of their differential algebras

     We relate the front and Lagrangian projections of grid number one knots in lens spaces, and we use this correspondence to determine the existence of augmentations of the Legendrian contact homology DGA.

     The Mathematics of Knots: Theory and Application (Computations in Mathematical and Computational Sciences), edited by Markus Banagl and Denis Vogel, Springer (2010) 143-168. The original publication is available at www.springerlink.com.

  1. Invariants for Legendrian knots in lens spaces

     Viewing lens spaces as quotients of circle-bundles, we construct a combinatorially-defined differential algebra whose homology is an invariant of the Legendrian isotopy class of K.

     Communications in Contemporary Mathematics, Vol. 13, No.1, 91-121.

  1. Constructing Seifert surfaces for n-bridge link projections

     This paper presents new algorithms for constructing Seifert surfaces for knots and links in the three-sphere.  We  apply the main algorithm to a knot whose canonical genus is less than its Seifert genus and show that the surface  constructed realizes the canonical genus.

     Journal of Knot Theory and Its Ramifications, Volume 19, Issue 2 (2010)

  1. The Thurston polytope for four-stranded pretzel links

     We use the Heegaard Floer link invariant to determine the Thurston polytopes of a members of a four-parameter family of pretzel links.  This family includes an infinite set of links with vanishing multivariable Alexander polynomial.

     Algebraic and Geometric Topology, 8 (2008)

My research focuses on knots and three-manifolds.  I am particularly interested in the Floer-theoretic tools of Heegaard Floer theory and Legendrian contact homology.  Both of these are defined via pseudo-holomorphic curves in symplectic manifolds, but they can be approached from a combinatorial perspective as well.  Generally speaking, the geometric theories are more powerful while the combinatorial versions are easier to work with.  I study the interplay between these two viewpoints, as well as their applications to topological and geometric questions.  In addition to contact geometry, I am also interested in topological topics such as Dehn surgery, Heegaard splittings, and link genus. 

Talks


To see a video of the talk I gave at MSRI in 2010, click here