Related papers: (1,1) almost L-space knots
We provide related Dehn surgery descriptions for rational homology spheres and a class of their regular finite cyclic covering spaces. As an application, we use the surgery descriptions to relate the Casson invariants of the covering spaces…
The trace of $n$-framed surgery on a knot in $S^3$ is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a manifold can be realised by a locally flat embedded 2-sphere…
We give an algorithm for computing the knot Floer homology of a $ (1,1) $ knot from a particular presentation of its fundamental group.
We show that the integer homology sphere obtained by splicing two nontrivial knot complements in integer homology sphere L-spaces has Heegaard Floer homology rank strictly greater than one. In particular, splicing the complements of…
There have been a number of constructions of Lagrangian Floer homology invariants for $3$-manifolds defined in terms of symplectic character varieties arising from Heegaard splittings. With the aim of establishing an Atiyah-Floer…
For each partial flag manifold of SU(N), we define a Floer homology theory for knots in 3-manifolds, using instantons with codimension-2 singularities.
We use the combinatorial techniques of graphs of intersection to study reducible Dehn surgeries on knots in the three-sphere. In particular, in the event that a reducible surgery on a knot K in the three-sphere of slope r produces a…
(Original version of PhD thesis, submitted in Spring 2009 to Harvard University. Provides a solution of the $p > k^2$ case, corresponding to Berge families I-VI, of the "Lens space realization problem" later solved in entirety by Greene.)…
We describe a simple formula for computing the Heegaard Floer multicurve invariant of double tangles from the Heegaard Floer multicurve invariant of knot complements. A comparison with a similar multicurve invariant for Conway tangles in…
Using the rational surgery formula for the Casson--Walker--Lescop invariant of links in the $3$-sphere, we show that any null-homologous knot in a rational homology sphere admits at most two pairs of integral purely cosmetic surgeries. We…
We establish a relationship between Heegaard Floer homology and the fractional Dehn twist coefficient of surface automorphisms. Specifically, we show that the rank of the Heegaard Floer homology of a 3-manifold bounds the absolute value of…
We consider manifold-knot pairs $(Y,K)$ where $Y$ is a homology sphere that bounds a homology ball. We show that the minimum genus of a PL surface $\Sigma$ in a homology ball $X$ such that $\partial (X, \Sigma) = (Y, K)$ can be arbitrarily…
A family of one-vertex triangulations of 3-manifolds, layered-triangulations, is defined. Layered-triangulations are first described for handlebodies and then extended to all 3-manifolds via Heegaard splittings. A complete and detailed…
The instanton Floer homology of a knot in the three-sphere is a vector space with a canonical mod 2 grading. It carries a distinguished endomorphism of even degree,arising from the 2-dimensional homology class represented by a Seifert…
In this paper we investigate the Alexander polynomial of (1,1)-knots, which are knots lying in a 3-manifold with genus one at most, admitting a particular decomposition. More precisely, we study the connections between the Alexander…
Ozsvath and Szabo conjectured that knot Floer homology detects fibred knots. We propose a strategy to approach this conjecture based on Gabai's theory of sutured manifold decomposition and contact topology. We implement this strategy for…
In a lens space X of order r a knot K representing an element of the fundamental group pi_1 X = Z/rZ of order s <= r contains a connected orientable surface S properly embedded in its exterior X-N(K) such that the boundary of S intersects…
Many three dimensional manifolds are two-fold branched covers of the three dimensional sphere. However, there are some that are not. This paper includes exposition about two-fold branched covers and many examples. It shows that there are…
Let Y(r) be the closed, oriented three-manifold obtained by performing rational r-surgery on the right-handed trefoil knot in the three-sphere. Using contact surgery and the Heegaard Floer contact invariants we construct positive, tight…
Given a transverse link in the standard contact 3-sphere, we study the contact manifold that arises as a branched double cover of the sphere. We give a contact surgery description of such manifolds, which allows to determine the Heegaard…