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On L-spaces and left-orderable fundamental groups
Examples suggest that there is a correspondence between L-spaces and three-manifolds whose fundamental groups cannot be left-ordered. In this paper we establish the equivalence of these conditions
Bordered Floer homology for manifolds with torus boundary via immersed curves
This paper gives a geometric interpretation of bordered Heegaard Floer homology for manifolds with torus boundary. If $M$ is such a manifold, we show that the type D structure
On the geography and botany of knot Floer homology
This paper explores two questions: (1) Which bigraded groups arise as the knot Floer homology of a knot in the three-sphere? (2) Given a knot, how many distinct knots share its Floer homology?
Graph manifolds, left-orderability and amalgamation
We show that every irreducible toroidal integer homology sphere graph manifold has a left-orderable fundamental group. This is established by way of a specialization of a result due to Bludov and
Left-orderable fundamental groups and Dehn surgery
There are various results that frame left-orderability of a group as a geometric property. Indeed, the fundamental group of a 3-manifold is left-orderable whenever the first Betti number is positive;
Heegaard Floer homology for manifolds with torus boundary: properties and examples
This is a companion paper to earlier work of the authors, which interprets the Heegaard Floer homology for a manifold with torus boundary in terms of immersed curves in a punctured torus. We prove a
Surgery obstructions from Khovanov homology
For a 3-manifold with torus boundary admitting an appropriate involution, we show that Khovanov homology provides obstructions to certain exceptional Dehn fillings. For example, given a strongly
A calculus for bordered Floer homology
We consider a class of manifolds with torus boundary admitting bordered Heegaard Floer homology of a particularly simple form, namely, the type D structure may be described graphically by a disjoint
L-spaces, taut foliations, and graph manifolds
If $Y$ is a closed orientable graph manifold, we show that $Y$ admits a coorientable taut foliation if and only if $Y$ is not an L-space. Combined with previous work of Boyer and Clay, this implies
On cabled knots, Dehn surgery, and left-orderable fundamental groups
Previous work of the authors establishes a criterion on the fundamental group of a knot complement that determines when Dehn surgery on the knot will have a fundamental group that is not