Related papers: Instantons and L-space surgeries

Ozsv\'ath and Szab\'o conjectured that knot Floer homology detects fibred knots in $S^3$. We will prove this conjecture for null-homologous knots in arbitrary closed 3--manifolds. Namely, if $K$ is a knot in a closed 3--manifold $Y$, $Y-K$…

We construct an infinite family of knots in rational homology spheres with irreducible, non-fibered complements, for which every non-longitudinal filling is an L-space.

A knot $K\subset S^3$ is called $SU(2)$-abundant if it satisfies two conditions: first, for all but finitely many $r\in\mathbb{Q}\backslash\{0\}$, there exists an irreducible representation $\pi_1(S^3_r(K))\to SU(2)$; second, any slope…

We prove an integral surgery formula for framed instanton homology $I^\sharp(Y_m(K))$ for any knot $K$ in a $3$-manifold $Y$ with $[K]=0\in H_1(Y;\mathbb{Q})$ and $m\neq 0$. Though the statement is similar to Ozsv\'ath-Szab\'o's integral…

For any $s \in [-\infty, 0] $ and oriented homology 3-sphere $Y$, we introduce a homology cobordism invariant $r_s(Y)\in (0,\infty]$. The values $\{r_s(Y)\}$ are included in the critical values of the $SU(2)$-Chern-Simons functional of $Y$,…

Li-Xie-Zhang classified instanton Floer minimal knots in balanced sutured manifolds subject to a condition on the fundamental group. In this paper, we give a similar classification in the Heegaard Floer homology setting. Since our…

We present two examples of strongly invertible L-space knots whose surgeries are never the double branched cover of a Khovanov thin link in the 3-sphere. Consequently, these knots provide counterexamples to a conjectural characterization of…

We show that if $Y$ is the boundary of an almost-rational plumbing, then the framed instanton Floer homology $\smash{I^\#(Y)}$ is isomorphic to the Heegaard Floer homology $\smash{\widehat{\mathit{HF}}(Y; \mathbb{C})}$. This class of…

The $L$-space conjecture asserts the equivalence, for prime 3-manifolds, of three properties: not being an $L$-space, having a left-orderable fundamental group, and admitting a co-oriented taut foliation. We investigate these properties for…

In this paper, we generalize the work of the second author and prove a grading shifting property, in sutured monopole and instanton Floer theories, for general balanced sutured manifolds. This result has a few consequences. First, we offer…

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.

Using instanton homology with coefficients in $Z/2$ we construct a homomorphism $q_2$ from the homology cobordism group in dimension 3 to the integers which is not a rational linear combination of the instanton $h$--invariant and the…

We prove that (1,1) non-L-space knots in $S^3$ and lens spaces are persistently foliar. This provides positive evidence for the L-space conjecture.

In a recent paper, the first author and his collaborator developed a method to compute an upper bound of the dimension of instanton Floer homology via Heegaard Diagrams of 3-manifolds. For a knot inside S3, we further develop an algorithm…

We show that if a positive integral surgery on a knot K inside a homology sphere X with Seifert genus g(K) results in an induced knot K_n in X_n(K)=Y which has simple Floer homology, we should have n>=2g(K). Moreover, if X is the standard…

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 knots in $S^3$ with infinitely many $SU(2)$-cyclic surgeries, which are Dehn surgeries such that every representation of the resulting fundamental group into $SU(2)$ has cyclic image. We show that for every such nontrivial knot…

Using instanton Floer theory, extending methods due to Froyshov, we determine the definite lattices that arise from smooth 4-manifolds bounded by certain homology 3-spheres. For example, we show that for +1 surgery on the (2,5) torus knot,…

Given any genus-two, hyperbolic, fibered knot in $S^3$ with nonzero fractional Dehn twist coefficient, we show that its pseudo-Anosov representative has a fixed point. Combined with recent work of Baldwin--Hu--Sivek, this proves that knot…

In this paper, we use Heegaard Floer homology to study reducible surgeries. In particular, suppose K is a non-cable knot in the three-sphere with an L-space surgery. If p-surgery on K is reducible, we show that p equals 2g(K)-1. This…