Related papers: Decomposing sutured monopole and Instanton Floer h…
We continue our study of the integer-valued knot invariants $\nu^\sharp(K)$ and $r_0(K)$, which together determine the dimensions of the framed instanton homologies of all nonzero Dehn surgeries on $K$. We first establish a "conjugation"…
We show that the knot group of any knot in any integer homology sphere admits a non-abelian representation into $SU(3)$ such that meridians are mapped to matrices whose eigenvalues are the three distinct third roots of unity. This answers…
For any link of two components in an integral homology sphere, we define an instanton Floer homology whose Euler characteristic is the linking number between the components of the link. We relate this Floer homology to the Kronheimer-Mrowka…
We define a homological action on sutured instanton Floer homology. This action is well-defined up to scalars, and behaves well under connected sums and sutured manifold decompositions. As an application, we show that instanton knot…
We associate several invariants to a knot in an integer homology 3-sphere using $SU(2)$ singular instanton gauge theory. There is a space of framed singular connections for such a knot, equipped with a circle action and an equivariant…
The cosmetic surgery conjecture predicts that for a non-trivial knot in the three-sphere, performing two different Dehn surgeries results in distinct oriented three-manifolds. Hanselman reduced the problem to $\pm 2$ or $\pm 1/n$ surgeries…
For closed 3-manifolds, Heegaard Floer homology is related to the Thurston norm through results due to Ozsv\'ath and Szab\'o, Ni, and Hedden. For example, given a closed 3-manifold Y, there is a bijection between vertices of the HF^+(Y)…
We construct a spectral sequence from the reduced odd Khovanov homology of a link converging to the framed instanton homology of the double cover branched over the link, with orientation reversed. Framed instanton homology counts certain…
We prove that instanton L-space knots are fibered and strongly quasipositive. Our proof differs conceptually from proofs of the analogous result in Heegaard Floer homology, and includes a new decomposition theorem for cobordism maps in…
In this paper, we define real link Floer homology for strongly invertible and doubly periodic links in closed real $3$-manifolds with connected fixed sets, which generalizes real Heegaard Floer homology and real sutured Heegaard Floer…
To a nullhomologous knot $K$ in a 3-manifold $Y$, knot Floer homology associates a bigraded chain complex over $\mathbb{F}[U,V]$ as well as a collection of flip maps; we show that this data can be interpretted as a collection of decorated…
Symplectic instanton homology is an invariant for closed oriented three-manifolds, defined by Manolescu and Woodward, which conjecturally corresponds to a symplectic version of a variant of Floer's instanton homology. In this thesis we…
This is a companion paper to earlier work of the authors, which proved an integral surgery formula for framed instanton homology. First, we present an enhancement of the large surgery formula, a rational surgery formula for null-homologous…
Knot Floer homology is a knot invariant defined using holomorphic curves. In more recent work, taking cues from bordered Floer homology,the authors described another knot invariant, called "bordered knot Floer homology", which has an…
This is the second paper in this series. Following the setup of Meng-Taubes, we define the monopole Floer homology for any pair $(Y,\omega)$, where $Y$ is a compact oriented 3-manifold with toroidal boundary and $\omega$ is a suitable…
It has been a central open problem in Heegaard Floer theory whether cobordisms of links induce homomorphisms on the associated link Floer homology groups. We provide an affirmative answer by introducing a natural notion of cobordism between…
We prove for the first time that knot Floer homology and Khovanov homology can detect non-fibered knots, and that HOMFLY homology detects infinitely many such knots; these theories were previously known to detect a mere six knots, all…
We study the instanton Floer homology for links in $\mathbb{RP}^3$ and compute the second page of Kronheimer--Mrowka's spectral sequence. As an application, we show that Khovanov homology detects the unknot and the projective unknot in…
In this paper we define instanton Floer homology groups for a pair consisting of a compact oriented 3-manifold with boundary and a Lagrangian submanifold of the moduli space of flat SU(2)-connections over the boundary. We carry out the…
We prove that Khovanov homology detects the trefoils. Our proof incorporates an array of ideas in Floer homology and contact geometry. It uses open books; the contact invariants we defined in the instanton Floer setting; a bypass exact…