Welcome. I'm in my fifth year as a grad student here at Columbia, graduating this May. My advisor is Peter Ozsvath, and my thesis is a compilation of several of the results below, on Heegaard Floer homology and open books. Next year I'll be at Princeton/IAS. Right now, I'm writing up my thesis. When that's over with, I'll post some more cool math stuff.
Tight contact structures and genus one fibered knots. Algebraic and Geometric Topology 7: 701--735 (2007).
We classify tightness for contact structures compatible with genus one, one boundary component open books given the monodromy. We also reveal a new infinite family of hyperbolic 3-manifolds with no taut foliations. In addition, we classify some of the L-spaces which arise as these open books, and more...
A note on genus one fibered knots in lens spaces We answer a question posed by Ken Baker, and find infinitely many lens spaces which don't contain any knot whose exterior is a once-punctured torus bundle.
Computations of Heegaard Floer knot homology (Joint with W. D. Gillam). Using a combinatorial approach described in a recent paper of Manolescu, Ozsvath, and Sarkar we compute the Heegaard-Floer knot homology of all knots with at most 12 crossings as well as the tau invariant for knots through 11 crossings. We review the basic constructions, giving two examples that can be worked out by hand, and explain some ideas we used to simplify the computation. We conclude with a discussion of knot Floer homology for small knots, closely examining the Kinoshita-Terasaka knot KT_2,1 and its Conway mutant.
For this paper, we wrote a C++ program which computes the hat version of HFK given a grid diagram. See below.
Comultiplicativity of the Ozsvath-Szabo contact invariant Mathematical Research Letters 15(2): 273--287 (2008).
Let S be a surface with boundary and suppose that g and h are diffeomorphisms of S which restrict to the identity on the boundary. Suppose further that the contact structures compatible with the open books (S,g) and (S,h) have non-vanishing Ozsvath-Szabo contact invariants. Then the contact structure compatible with the open book (S,hg) has non-vanishing Ozsvath-Szabo contact invariant as well. This follows from a naturality feature of the contact invariant under a certain comultiplication map on Heegaard Floer homology. We extend this naturality to HF^+, obtaining obstructions to the compatibility of contact structures with planar open books.
Heegaard Floer homology and genus one, one boundary component open books (This used to be called ``Heegaard Floer homology and branched double covers of 3-braids.") We compute the Heegaard Floer homology of any rational homology 3-sphere with a genus one, one boundary component open book decomposition. In addition, we compute the Heegaard Floer homology of any torus bundle over the circle with first Betti number equal to one, and we compare our results with those of Lebow on the embedded contact homology of such torus bundles. We use these computations to narrow down somewhat the class of 3-braid knots with finite concordance order, and to identify all quasi-alternating links with braid index at most 3.
I wrote a C++ program to go with this paper which computes various things related to 3-braids and open books. See below.
Programs
HFK This program computes the hat version of knot Floer homology given a grid diagram for a knot.
Trans This program determines whether Plamenevskaya's invariant of transverse braids vanishes in reduced Khovanov homology.
3Braid This program decides whether a given 3-braid is quasi-alternating, computes HF of its branched double cover, determines whether the corresponding contact structure is tight, etc.
Older Research
A reciprocity theorem for monomer-dimer coverings (Joint with N. Anzalone, I. Bronshtein, and T. K. Petersen). We study the combinatorics of monomer-dimer coverings of an m by n rectangular grid, finding combinatorial interpretations when n<0.
Self-conjugate t-core partitions, sums of squares, and p-blocks of A_n Journal of Algebra297 (2006), 438-452.
(Joint with M. Depweg, B. Ford, A. Kunin, L. Sze). We prove that if t is an integer with t=8 or t>=10, then every integer n > 2 has a self-conjugate t-core partition. This result has consequences in the representation theory of alternating groups, and has a version as a theorem about the representation of integers by sums of squares. We also give an infinite sequence of integers that have no self-conjugate 9-core partitions.
