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Research
Heegaard Floer theory is a set of invariants of three- and four-dimensional manifolds which
have significantly impacted the study of many areas of low dimensional topology including Dehn
surgery and foliation theory. To date, my research has focused primarily on the interactions between knot theory
and various Floer homology invariants. I am currently writing a paper joint with Jen Hom
and Tye Lidman about a new L-space satellite operation using Berge-Gabai knots as the pattern.
Here is my Research Statement.
Papers
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Berge-Gabai knots and L-space satellite operations [Abstract]
with Jen Hom and Tye Lidman
In preparation.
Let $P(K)$ be a satellite knot where the pattern, $P$, is a Berge-Gabai knot (i.e. a
knot in the solid torus with a non-trivial solid torus Dehn surgery), and the companion, $K$, is
an arbitrary knot in S3. We prove that $P(K)$ is an L-space knot if and only if $K$ is an L-space
knot and a numerical condition relating the 1-bridge presentation of $P$ to the genus of $K$ is
satified. This generalizes the result for cables due to Hedden and Hom.
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On the knot Floer homology of twisted torus knots [Abstract]
Submitted to the journal of Int. Math. Res. Not. IMRN, 2013.
In this paper we study the knot Floer homology of a subfamily of twisted $(p, q)$
torus knots where $q$ \equiv \pm 1 (mod $p$). Specifically, we classify the knots in this subfamily that
admit L-space surgeries. To do calculations, we use the fact that these knots are (1,1) knots
and, therefore, admit a genus one Heegaard diagram.
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Seifert surfaces distinguished by sutured Floer
homology but not its Euler characteristic
[Abstract]
Accepted with revisions in the journal of Topology and its Applications, 2013.
In this paper we find a family of knots with trivial Alexander polynomial, and
construct two non-isotopic Seifert surfaces for each member in our family. In order to distinguish
the surfaces we study the sutured Floer homology invariants of the sutured manifolds obtained
by cutting the knot complements along the Seifert surfaces. Our examples provide the first
use of sutured Floer homology, and not merely its Euler characteristic (a classical torsion), to
distinguish Seifert surfaces. Our technique uses a version of Floer homology, called ``longitude
Floer homology" in a way that enables us to bypass the computations related to the $SFH$ of
the complement of a Seifert surface.
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Selected Talks
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Heegaard Floer homology and L-space knots
Joint Mathematics Meetings, January 2014
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On the knot Floer homology of twisted torus knots
Topology Seminar, Michigan State University, September 2013
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Heegaard Floer homology and L-space knots
Short Talks, Graduate Workshop on Topology and Invariants of 4-manifolds, University of Minnesota, August 2013
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Seifert surfaces distinguished by sutured Floer homology
Geometry Seminar, University of Virginia, October 2012
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Sutured Floer homology distinguishes between two Seifert surfaces but not Turaev torsion [Beamer Presentation]
Tenth Annual Graduate Student Topology and Geometry Conference,
Indiana University, April 2012
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