Jennifer Hom

The knot Floer complex and the smooth concordance group

We define a new smooth concordance homomorphism based on the knot Floer complex and an associated concordance invariant, epsilon. As an application, we show that an infinite family of topologically slice knots are independent in the smooth concordance group.

A survey on Heegaard Floer homology and concordance

In this survey article, we discuss several different knot concordance invariants coming from the Heegaard Floer homology package of Ozsvath and Szabo. Along the way, we prove that if two knots are concordant, then their knot Floer complexes satisfy a certain type of stable equivalence.

ON THE KNOT FLOER FILTRATION OF THE CONCORDANCE GROUP

The knot Floer complex together with the associated concordance invariant e can be used to define a filtration on the smooth concordance group. We show that the indexing set of this filtration contains ℕ × ℤ as an ordered subset.

Berge-Gabai knots and L-space satellite operations

Let P(K) be a satellite knot where the pattern P is a Berge–Gabai knot (ie a knot in the solid torus with a nontrivial solid torus Dehn surgery) and the companion K is a nontrivial knot in S^3. We prove that P(K) is an L–space knot if and only if K is an L–space knot and P is sufficiently positively twisted relative to the genus of K. This generalizes the result for cables due to Hedden [Int. Math. Res. Not. 2009 (2009) 2248–2274] and Hom [Algebr. Geom. Topol. 11 (2011) 219–223].

Satellite knots and L-space surgeries

We give sufficient conditions for a satellite knot to admit an L-space surgery, and use this result to give new infinite families of patterns which produce satellite L-space knots.

Involutive Heegaard Floer homology and rational cuspidal curves: RATIONAL CUSPIDAL CURVES

We use invariants of Hendricks and Manolescu coming from involutive Heegaard Floer theory to find constraints on possible configurations of singular points of a rational cuspidal curve of odd degree in the projective plane. We show that the results do not carry over to rational cuspidal curves of even degree.