Corrigendum to "Knot Floer homology detects fibred knots"
Yi Ni
纠正了\cite{Ni}中引用JSJ理论的一个错误。\cite{Ni}中的一些参数也相应地进行了轻微修改。
We correct a mistake on the citation of JSJ theory in \cite{Ni}. Some arguments in \cite{Ni} are also slightly modified accordingly.
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Knot Floer homology detects fibred knots
Ozsváth和Szabó证实，结弗洛尔同调检测到$S^3$中的重复结。我们将证明这一猜想的零同源纽结在任意封闭的3-流形。也就是说，如果$K$是闭3-流形$Y$中的纽结，$Y-K$是不可约的，并且$\hat{HFK}(Y,K)$是幺半群，则$K$是可约的。证明依赖于Gabai，Ozsváth-Szabó，Ghiggini和作者以前的作品。一个推论是，如果$S^3$中的一个结允许透镜空间手术，则该结被切除。
Heegaard Floer homology and fibred 3--manifolds
给定一个闭的3-流形$Y$，我们证明了Heegaard弗洛尔同调决定了$Y$是否在一个具有负欧拉示性数的圆上形成纤维.这是一个类似的早期结果结证明了Ghiggini和作者。
We prove that the Knot Floer homology group of a fibred knot of genus g in the Alexander grading 1-g is isomorphic to a version of the fixed point Floer homology of an area-preserving representative of the monodromy.
Knot Floer homology detects genus-one fibred knots
Ozsvath和Szabo证实，结弗洛尔同调检测到了非对称结。我们提出了一个策略，以接近这个猜想的基础上Gabai的缝合流形分解和接触拓扑理论。我们实施这一战略的属一结，获得作为推论，如果合理的手术结$K$的庞加莱同源球$Σ(2,3,5)$，然后$K$是左手三叶结。
Closed 3-braids are nearly fibred
Ozsváth和Szabó指出，结弗洛尔同调检测到了重复的连接。我们将对闭3-辫证明这个猜想，通过分类闭3-辫。特别是，给定一个非平凡的闭3-辫，它要么是双绞的，要么它与双绞的链环相差半捻。证明使用Gabai的圆盘分解方法。
We show that the properties of admitting a co-oriented taut foliation and having a left-orderable fundamental group are equivalent for rational homology $3$-sphere graph manifolds and relate them to the property of not being a Heegaard-Floer L-space. This is accomplished in several steps. First we show how to detect families of slopes on the boundary of a Seifert fibred manifold in four different fashions - using representations, using left-orders, using foliations, and using Heegaard-Floer homology. Then we show that each method of detection determines the same family of detected slopes. Next…
Heegaard Floer homology of broken fibrations over the circle
我们将Perutz的拉格朗日匹配不变量推广到不需要使用全纯面组技术来覆盖的三维流形上。对于某些极端自旋^c结构，我们证明了这些不变量与Ozsvath-Szabo的Heegaard Floer不变量的同构。作为应用，我们给出了某些3-流形类的Heegaard弗洛尔同调的新计算，以及Juhasz缝合弗洛尔同调的一个特征.
We write down an explicit formula for the $+$ version of the Heegaard Floer homology (as an absolutely graded vector space over an arbitrary field) of the results of Dehn surgery on a knot $K$ in $S^3$ in terms of homological data derived from $CFK^{\infty}(K)$. This allows us to prove some results about Dehn surgery on knots in $S^3$. In particular, we show that for a fixed manifold there are only finitely many alternating knots that can produce it by surgery. This is an improvement on a recent result by Lackenby and Purcell. We also derive a lower bound on the genus of knots depending on the…
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 an algorithm that computes the Floer homologies of a family of sutured handle-bodies. Second, we obtain a canonical decomposition of sutured monopole and instanton Floer homologies and build polytopes for these two theories, which was initially achieved by Juhász for sutured (Heegaard) Floer homology. Third, we establish a Thurston-norm detection result for …
Heegaard Floer homology and maximal twisting numbers
本文通过调整Ozsváth-Szabó的完整路径算法，建立了Seifert纤维空间上负扭转紧致接触结构与Heegaard Floer同调群之间的对应关系，揭示了这些结构的分类及其接触不变量c^+的区分。
The sutured Floer homology polytope
本文推广了作者提出的缝合弗洛尔同调理论。首先证明了一个附加不等式，然后在H ^2（M，\partial M; R）中定义了一个由支持非零弗洛尔同调群的Spin^c-结构所张成的多面体P（M，g）.若（M，g）->（M '，g'）是紧曲面分解，则自然映射将P（M '，g'）投影到P（M，g）的一个面上，若H_2（M）= 0，则P（M，g）的每一个面都可以通过这种方法得到。证明了如果（M，g）是约化的，水平素的，且H_2（M）= 0，则P（M，g）是H^2（M，\partial M; R）中的最大维数.这意味着如果rk（SFH（M，g））< 2^{k+1}，则（M，g）的深度至多为2k。此外，SFH作为一个复杂性的平衡缝合流形。特别是，结弗洛尔同调的顶项的秩限制了结补的拓扑复杂性，除了简单地检测被隐藏的结之外。
Floer homology and surface decompositions
缝合弗洛尔同调是作者以前定义的平衡缝合流形的一个不变量，记作SFH。在本文中，我们给出了一个公式，表明如何在表面分解下，这个不变量的变化。特别地，如果（M，γ）-->（M '，M'）是缝合流形分解，则SFH（M '，M'）是SFH（M，γ）的直和项。为了证明分解公式，我们给出了一个从定义（M，γ）的平衡图计算SFH（M，γ）的算法，该算法推广了Sarkar和Wang的算法。作为推论，我们得到，如果（M，γ）是紧的，则SFH（M，γ）非零。其他应用包括简单的证明结果Ozsvath和Szabo的链接弗洛尔同调检测瑟斯顿规范，和一个定理的镍结弗洛尔同调检测blackred结。我们的证明不使用任何接触几何。利用这些方法还证明了：若K是有理同调三维球面Y中的亏格g纽结，且其Alexander多项式的首项系数ag不为零，且若\hat{HFK}（Y，K，g）的秩< 4，则纽结补存在一个深度< 2的紧叶理横截于N（K）的边界.
We show that a decorated knot concordance $C$ from $K$ to $K'$ induces a homomorphism $F_C$ on knot Floer homology that preserves the Alexander and Maslov gradings. Furthermore, it induces a morphism of the spectral sequences to $\widehat{HF}(S^3) \cong \mathbb{Z}_2$ that agrees with $F_C$ on the $E^1$ page and is the identity on the $E^\infty$ page. It follows that $F_C$ is non-vanishing on $\widehat{HFK}_0(K, τ(K))$. We also obtain an invariant of slice disks in homology 4-balls bounding $S^3$. If $C$ is invertible, then $F_C$ is injective, hence $\dim \widehat{HFK}_j(K,i) \le \dim \widehat{…
A spectral sequence for Lagrangian Floer homology
证明了在多重Dehn扭曲下，存在一个收敛于拉格朗日子流形象的弗洛尔同调的Lagrange弗洛尔同调的谱序列.序列的$E_1$项由所涉及的德恩扭曲的“分辨率”的超立方体给出。证明依赖于由于Wehrheim和Woodward的纤维Dehn扭曲的确切三角形。作为应用，我们得到了一个从Khovanov同调到辛Khovanov同调的谱序列.当一个3-流形$M$是由两个曲面体通过一个曲面同态$ϕ$粘合而得到时，我们得到了一个收敛于$M$的Heegaard-弗洛尔同调的谱序列，其$E_1$项是一个超立方体，它是通过分解$ϕ$中的Dehn扭曲的不同方法得到的.后一个序列将分支双覆盖的谱序列推广到一般的闭3-流形（即那些不是链的分支双覆盖的流形），但是它的$E_2$项不是3-流形不变量。这给出了闭3-流形的HF-帽的秩的上界。
We investigate the line between tight and overtwisted for surgeries on fibred transverse knots in contact 3-manifolds. When the contact structure $ξ_K$ is supported by the fibred knot $K \subset M$, we obtain a characterisation of when negative surgeries result in a contact structure with non-vanishing Heegaard Floer contact class. To do this, we leverage information about the contact structure $ξ_{\overline{K}}$ supported by the mirror knot $\overline{K} \subset -M$. We derive several corollaries about the existence of tight contact structures, L-space knots outside $S^3$, non-planar contact …
A knot \widetilde{K} \subset S^3 is q-periodic if there is a \mathbb Z_q-action preserving \widetilde{K} whose fixed set is an unknot U. The quotient of \widetilde{K} under the action is a second knot K. We construct equivariant Heegaard diagrams for q-periodic knots, and show that Murasugi's classical condition on the Alexander polynomials of periodic knots is a quick consequence of these diagrams. For \widetilde{K} a two-periodic knot, we show there is a spectral sequence whose E^1 page is \hat{\mathit{HFL}}(S^3,\widetilde{K}\cup U)\otimes V^{\otimes (2n-1)})\otimes \mathbb Z_2((θ)) and whos…
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