Related papers: Rank three instantons, representations and sutures

We prove new adjunction inequalities for embedded surfaces in four-manifolds with non-negative self-intersection number by using the Donaldson invariants. These formulas are completely analogous to the ones obtained by Ozsv\'ath and Szab\'o…

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…

A. Casson defined an intersection number invariant which can be roughly thought of as the number of conjugacy classes of irreducible representations of $\pi_1(Y)$ into $SU(2)$ counted with signs, where $Y$ is an oriented integral homology…

In this paper we extend the idea of bordered Floer homology to knots and links in $S^3$: Using a specific Heegaard diagram, we construct gluable combinatorial invariants of tangles in $S^3$, $D^3$ and $I\times S^2$. The special case of…

Using Heegaard Floer homology, we construct a numerical invariant for any smooth, oriented $4$-manifold $X$ with the homology of $S^1 \times S^3$. Specifically, we show that for any smoothly embedded $3$-manifold $Y$ representing a…

We construct an obstruction for the existence of embeddings of homology $3$-sphere into homology $S^3\times S^1$ under some cohomological condition. The obstruction is defined as an element in the filtered version of the instanton Floer…

We compute the involutive Heegaard Floer homology of the family of three-manifolds obtained by plumbings along almost-rational graphs. (This includes all Seifert fibered homology spheres.) We also study the involutive Heegaard Floer…

This paper studies the existence of $2$-torsion in instanton Floer homology with $\mathbb{Z}$ coefficients for closed $3$-manifolds and singular knots. First, we show that the non-existence of $2$-torsion in the framed instanton Floer…

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…

In an earlier paper, we introduced a knot invariant for a null-homologous knot K in an oriented three-manifold Y, which is closely related to the Heegaard Floer homology of Y. In this paper we investigate some properties of these knot…

Conjecturally, a knot in the 3-sphere has only finitely many non-integer non-characterizing slopes. We verify this conjecture for all knots with knot Floer homology satisfying certain simplicity conditions. The class of knots satisfying our…

Let $G$ be a nonabelian, simple group with a nontrivial conjugacy class $C \subseteq G$. Let $K$ be a diagram of an oriented knot in $S^3$, thought of as computational input. We show that for each such $G$ and $C$, the problem of counting…

Surgery exact triangles in various 3-manifold Floer homology theories provide an important tool in studying and computing the relevant Floer homology groups. These exact triangles relate the invariants of 3-manifolds, obtained by three…

We exhibit the first example of a knot in the three-sphere with a pair of minimal genus Seifert surfaces that can be distinguished using the sutured Floer homology of their complementary manifolds together with the Spin^c-grading. This…

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…

We introduce explicit holonomy perturbations of the Chern-Simons functional on a 3-ball containing a pair of unknotted arcs. These perturbations give us a concrete local method for making the moduli spaces of flat singular SO(3) connections…

Kronheimer and Mrowka defined invariants of balanced sutured manifolds using monopole and instanton Floer homology. Their invariants assign isomorphism classes of modules to balanced sutured manifolds. In this paper, we introduce…

Meier and Zupan showed that every surface in the four-sphere admits a bridge trisection and can therefore be represented by three simple tangles. This raises the possibility of applying methods from link homology to knotted surfaces. We use…

We consider the question of when a rational homology 3-sphere is rational homology cobordant to a connected sum of lens spaces. We prove that every rational homology cobordism class in the subgroup generated by lens spaces is represented by…

This is a companion paper to an earlier work of the authors. In this paper, we provide an axiomatic definition of Floer homology for balanced sutured manifolds and prove that the graded Euler characteristic $\chi_{\rm gr}$ of this homology…