Benjamin Audoux
We define a grid presentation for singular links i.e. links with a finite number of rigid transverse double points. Then we use it to generalize link Floer homology to singular links. Besides the consistency of its definition, we prove that this homology is acyclic under some conditions which naturally make its Euler characteristic vanish.
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These are the lecture notes for a course on Heegaard Floer homology held at PCMI in Summer 2019. We describe Heegaard diagrams, Heegaard Floer homology, knot Floer homology, and the relationship between the knot and 3-manifold invariants.
We review some recent results in knot concordance and homology cobordism. The proofs rely on various forms of Heegaard Floer homology. We also discuss related open problems.
A well-known conjecture of Rasmussen states that for any knot $K$ in $S^{3}$, the rank of the reduced Khovanov homology of $K$ is greater than or equal to the rank of the reduced knot Floer homology of $K$. This rank inequality is supposed to arise as the result of a spectral sequence from Khovanov homology to knot Floer homology. Using an oriented cube of resolutions construction for a homology theory related to knot Floer homology, we prove this conjecture.
The symplectic Floer homology of the figure eight knot
本文计算了8字形纽结的辛弗洛尔同调。这提供了具有平凡辛弗洛尔同调的第一非平凡纽结。
Knot Floer homology, link Floer homology and link detection
受Khovanov同调启发，本文证明了knot和link Floer同调能检测多种特定链环，发现了无限多对具有相同Khovanov与knot Floer同调但可被link Floer同调区分的链环，并给出了annular Khovanov同调的拓扑应用。
We prove that the rank of knot Floer homology detects the Hopf links, and generalize this result further to classify the links of the second smallest knot Floer homology. We also prove a knot Floer homology analog of arXiv:1910.04246v1 [math.GT] and give a partial answer to when the equality holds in the rank inequality between the knot Floer homology of a link and its sublinks.
Instanton Floer homology for two-component links
对于整同调球中两个分量的任何链环，我们定义了一个瞬子弗洛尔同调，它的欧拉示性数是链环中两个分量之间的环绕数.我们把这种弗洛尔同调与纽结的克朗海默-姆洛卡瞬子弗洛尔同调联系起来。我们还表明，在3-球面的两个组件的链接，弗洛尔同调不消失，除非链接是分裂的。
Inspired by work of Lipshitz-Sarkar, we show that the module structure on link Floer homology detects split links. Using results of Ni, Alishahi-Lipshitz, and Lipshitz-Sarkar, we establish an analogous detection result for sutured Floer homology.
Localization and the Floer homology of strongly invertible knots
该研究在定向强可逆纽结K的纽结Floer同调中建立两个谱序列，分别关联K与三维球面的Heegaard Floer同调、K的奇异纽结Floer同调与K的商纽结Floer同调。
This note corrects an erroneous statement in Lemma 3.8 of the author's paper Embedded Contact Homology and Seiberg-Witten Floer Homology IV which was published in Volume 14 of Geometry and Topology in 2009.
Framed Floer Homology
对于在整同调球面中表示为框架连接（L，Λ）上的外科手术的任何三流形，Manolescu和Ozsváth构造了一个链复形的超立方体，其同调计算了Y上Λ-框架外科手术的Heegaard弗洛尔同调。这进行了一个自然的过滤，存在于任何超立方体的链复形，我们研究的E_2页的相关谱序列，称为框架弗洛尔同调。本文的目的之一是证明框架弗洛尔同调是定向框架链环的不变量，而不是surgered流形的不变量。我们将讨论如何这涉及到一个尝试在组合证明的不变性的Heegaard弗洛尔同调。我们还表明，框架弗洛尔同调满足一个外科手术确切的三角形类似于Heegaard的弗洛尔同调。这种设置导致一个完全组合的Heegaard弗洛尔同调中的外科手术精确三角形的构造。最后，我们研究了两个分量链的框架弗洛尔同调，这些链激增到S^2 x S^1 # S^2 x S^1并且具有性质2 R。
Floer homology of Brieskorn homology spheres: solution to Atiyah's problem
本文回答了M.~ Atiyah，并给出了Brieskorn同调球面的弗洛尔同调在3-球面上的分支集的显式表达式.进一步证明了弗洛尔同调与纽结和3-流形的其它不变量的关系，其中包括W.~的$\barμ$ -不变量Neumann和L.~ Siebenmann和Jones多项式在证明我们自己的$ν$ -不变量的同调配边不变性方面取得了实质性的进展。
We prove a basic inequality for the d-invariants of a splice of knots in homology spheres. As a result, we are able to prove a new relation on the rank of reduced Floer homology under maps between Seifert fibered homology spheres, improving results of the first and second authors. As a corollary, a degree one map between two aspherical Seifert homology spheres is homotopic to a homeomorphism if and only if the Heegaard Floer homologies are isomorphic.
Real Monopole Floer Homology and Skein Exact Triangles
本文证明了实单极Floer同调的无定向skein正合三角形，并引入了Frøyshov型不变量。
We show that for a tangle $T$ with $-\partial^0T \cong \partial^1 T$ the Hochschild homology of the tangle Floer homology $\widetilde{\mathit{CT}}(T)$ is equivalent to the link Floer homology of the closure $T' = T/(-\partial^0T \sim \partial^1 T)$ of the tangle, linked with the tangle axis. In addition, we show that the action of the braid group on tangle Floer homology is faithful.
We construct cobordism maps on link Floer homology associated to decorated link cobordisms. The maps are defined on a curved chain homotopy type invariant. We describe the construction, and prove invariance. We also make a comparison with the graph TQFT for Heegaard Floer homology.
We use Heegaard Floer homology to define an invariant of homology cobordism. This invariant is isomorphic to a summand of the reduced Heegaard Floer homology of a rational homology sphere equipped with a spin structure and is analogous to Stoffregen's connected Seiberg-Witten Floer homology. We use this invariant to study the structure of the homology cobordism group and, along the way, compute the involutive correction terms for certain families of three-manifolds.
Floer homology and singular knots
我们定义弗洛尔同调理论的定向，奇异纽结在S^3，并表明，这些理论之一，可以定义组合为平面奇异纽结。
Heegaard-Floer Homology and a family of Brieskorn spheres
本文利用P. Ozsvath和Z. Szabo
Khovanov homology and the Involutive Heegaard Floer homology of branched double covers
本文利用 involutive Heegaard Floer 同调，借助 bordered Floer 同调重构与手术正合三角，推广了 Ozsváth-Szabó 分支双覆盖谱序列，建立 Khovanov 同调与 Heegaard Floer 同调间的联系。
Effective computation of knot Floer homology
我们扩展了Beliakova的方法计算结弗洛尔同调，并在一个公开的计算机程序中实现它。我们回顾了主要的编程和最优化方法使用。我们的程序，然后用来检查的弗洛尔同调的一个素数非交替结与小于12交叉没有扭转。
A higher-order genus invariant and knot Floer homology
众所周知，纽结弗洛尔同调检测纽结的亏格和亚历山大多项式。我们调查是否结弗洛尔同调的$K$检测到更多的结构的最小亏格塞弗特表面的$K$。我们定义了一个代数切片，亏格为1的纽结的不变量，并举例说明纽结的弗洛尔同调不能检测到这个不变量。最后，我们注意到某些元阿贝尔$L^2$ -签名从下面绑定这个不变量。
The contact invariant in sutured Floer homology
本文描述了具有凸边界的切触3-流形的一个不变量作为Juhász缝合弗洛尔同调的一个元素。我们的不变量推广了Heegaard弗洛尔同调中的接触不变量，在封闭的情况下，由于Ozsváth和Szabó。这个版本有一些澄清和新的数字。
In this paper we construct gluing maps and cobordism maps for sutured monopole Floer homology.
A concordance invariant from the Floer homology of +/- 1 surgeries
讨论了由Heegaard弗洛尔同调“校正项”和三球面纽结上的+/-1运算构造的协调不变量。
Floer lasagna modules from link Floer homology
该研究受链环千层面模构造启发，引入弗洛尔千层面模概念，用纽结弗洛尔同调替代Khovanov-Rozansky同调，描述了特定4-流形的该模并计算了若干例子。
We prove that if two knots are concordant, their involutive knot Floer complexes satisfy a certain type of stable equivalence.
We prove that the map on knot Floer homology induced by a strongly homotopy-ribbon concordance is injective.
The symplectic Floer homology of composite knots
我们发展了一种计算复合纽结的辛弗洛尔同调的方法。在\cite{li}中定义的纽结的辛弗洛尔同调自然地允许整数分次提升，它形成了一个滤除和诱导的谱序列.这样的谱序列收敛于\cite{li}中纽结的辛同调。我们证明了对于由辫子表示的复合结点，存在另一个谱序列收敛于$\Z$ -分次辛弗洛尔同调。
We compute bordered Floer homology CFDD of (2,2n)-torus link complement, and discuss assorted examples and type-DD structure homotopy equivalence.
Link Floer homology detects the Thurston norm
我们证明了，对于一个链接$L$在一个合理的同源3-球，链接弗洛尔同调检测瑟斯顿规范的补充。这推广了以前的结果由于Ozsváth，Szabó和作者。
In a recent paper, Lin, Ruberman and Saveliev proved a splitting formula expressing the Seiberg-Witten invariant $λ_{SW}(X)$ of a smooth $4$-manifold with rational homology of $S^1\times S^3$ in terms of the Frøyshov invariant $h(X)$ and a Lefschetz number in reduced monopole Floer homology. In this note we observe that a similar splitting formula holds in reduced instanton Floer homology.
Floer homology, clasp-braids and detection results
本文拓展Martin的辫轴检测结果，基于链环Floer同调给出近辫状链环的拓扑刻画，受近纤维化纽结研究启发，得到Khovanov同调等的多类检测应用。
Combinatorial Heegaard Floer homology and nice Heegaard diagrams
我们考虑3-流形Y的hat Heegaard弗洛尔同调的稳定化形式（即闭3-流形的Heegaard弗洛尔同调的U=0变体）。我们给出了一个组合算法来构造这个不变量，从Heegaard分解Y，并给出了其不变性的组合证明。
In this paper, we explore the interplay between contact structures and sutured monopole Floer homology. First, we study the behavior of contact elements, which were defined by Baldwin and Sivek, under the operation of performing Floer excisions, which was introduced to the context of sutured monopole Floer homology by Kronheimer and Mrowka. We then compute the sutured monopole Floer homology of some special balanced sutured manifolds, using tools closely related to contact geometry. For application, we obtain an exact triangle for the oriented skein relation in monopole theory and derive a con…
We construct maps on hat Heegaard Floer homology for cobordisms decorated with graphs. The graph TQFT allows for cobordisms with disconnected ends. Our construction uses Juhász's sutured Floer TQFT. We compute the maps for several elementary graph cobordisms. As an application, we compute the action of the fundamental group on hat Heegaard Floer homology.
Combinatorial Heegaard Floer homology and sign assignments
我们提供了一个完整的提升的组合定义的Heegaard弗洛尔同调漂亮的图，并表明，独立性的证明，使用方便的图适应这种设置。
This paper is a short introduction to the combinatorial version of tangle Floer homology defined in "Combinatorial tangle Floer homology". There are two equivalent definitions---one in terms of strand diagrams, and one in terms of bordered grid diagrams. We present both, discuss the correspondence, and carry out some explicit computations.
Degree-1 maps and rank inequalities in Heegaard Floer homology
本文利用bordered Floer同调的浸入曲线解释，证明了特定1度映射在Heegaard Floer同调中产生的秩不等式，为Heegaard Floer同调与instanton Floer同调关系的猜想提供了进一步证据。
Rank inequalities for the Heegaard Floer homology of Seifert homology spheres
建立了Seifert纤维积分同调球的Heegaard弗洛尔同调的约化味的秩不等式.将这些不等式与Seifert纤维空间之间的非零次映射的已知分类相结合，我们证明了Seifert同调球面之间的从Y'到Y的映射f产生如下不等式：|华氏度|秩HFred（Y）至多是秩HFred（Y '）。这些不等式也适用于结合算法的Nemethi给出一种方法来解决植物学问题的Heegaard弗洛尔同调的这些流形。
Continuing our previous work, we effectively compute connected Heegaard Floer homologies of two families of Brieskorn spheres realized as the boundaries of almost simple linear graphs. Using Floer theoretic invariants of Dai, Hom, Stoffregen, and Truong, we show that these Brieskorn spheres also generate infinite rank summands in the homology cobordism group. Our computations also have applications to the concordance of classical knots and $0$-concordance of $2$-knots.
Bordered Floer homology with integral coefficients for manifolds with torus boundary
本文为带环面边界的流形定义了整系数边界的Floer理论，恢复了Ozsváth、Stipsicz和Szabó的组合Heegaard Floer同调。
Knot Floer homology in cyclic branched covers
本文引入了S ^3中纽结K的一个不变量序列：K上m重循环分枝覆盖中原像的纽结弗洛尔同调群.在m重分支复盖是有理同调球面的情况下，我们将m重分支复盖中的纽结弗洛尔同调作为Turaev挠的一个倍数的范畴化.此外，当K是2-桥纽结时，我们证明了分支双覆盖中特定Spin^c结构中提升纽结的纽结弗洛尔同调与S^3中原始纽结K的纽结弗洛尔同调匹配。最后，我们给出了一个涉及S^3中两个具有相同纽结弗洛尔同调的纽结的计算，其中双分支覆盖中的纽结弗洛尔同调群不同于Z_2-分次群。
We compute the connected Heegaard Floer homology (defined by Hendricks, Hom, and Lidman) for a large class of 3-manifolds, including all linear combinations of Seifert fibered homology spheres. We show that for such manifolds, the connected Floer homology completely determines the local equivalence class of the associated $ι$-complex. Some identities relating the rank of the connected Floer homology to the Rokhlin invariant and the Neumann-Siebenmann invariant are also derived. Our computations are based on combinatorial models inspired by the work of Némethi on lattice homology.
We give new obstructions to the module structures arising in Heegaard Floer homology. As a corollary, we characterize the possible modules arising as the Heegaard Floer homology of an integer homology sphere with one-dimensional reduced Floer homology. Up to absolute grading shifts, there are only two. We use this corollary to show that the chain complex depicted by Ozsváth, Stipsicz, and Szabó to argue that there is no algebraic obstruction to the existence of knots with trivial $ε$ invariant and non-trivial $Υ$ invariant cannot be realized as the knot Floer complex of a knot.
Spectral sequences in unoriented link Floer homology
本文通过迭代修改的无向skein正合三角形，在二元域上构造了从多种Khovanov同调到多种无向链环Floer同调的谱序列，并给出$S^{3}$中纽结的镜像约化Khovanov同调到纽结Floer同调的谱序列。
We make use of link Floer homology to study cobordisms between links embedded in 4-dimensional ribbon homology cobordisms. Combining results of Daemi--Lidman--Vela-Vick--Wong and Zemke, we show that ribbon homology concordances induce split injections on $\mathcal{HFL}^-$. We also make use of the torsion submodule of $\text{HFL}^-$ to give restrictions on the number of critical points in ribbon homology concordances.
On rational blow-downs in Heegaard-Floer homology
受L. P. Roberts关于Heegaard-弗洛尔同调中有理blow-downs的结果的启发，我们研究了沿着3-流形的这种操作，这些操作作为$S^{3}$沿着几个非交替切片结点的分支双覆盖而出现。
Cable Links, Annuli and Sutured Floer homology
本文利用缝合 Floer 同调研究外部含环带的链环，刻画了在大参数 $m$ 下与特定 $L$-space 纽结缆链具有相同 Floer 同调的链环，并得到新的链环检测结果。
Floer homology beyond borders
本文概述了带边3-流形的Bordered Floer homology理论及其对闭3-流形Heegaard Floer homology的扩展，重点介绍了近期的计算工具发展与应用。
Heegaard Floer Homology and Triple Cup Products
我们给出了所有三维流形和扭转Spin^c结构的模系数为2的Heegaard弗洛尔同调的无穷味的完整计算。计算结果与Ozsvath和Szabo的修正计算结果一致。因此，这建立了与马克杯模2同调的同构。
Khovanov-Floer theories are a class of homological link invariants which admit spectral sequences from Khovanov homology. They include Khovanov homology, Szab{ó}'s geometric link homology, singular instanton homology, and various Floer theories applied to branched double covers. In this short note we show that certain strong Khovanov-Floer theories, including Szab{ó} homology and singular instanton homology, are invariant under Conway mutation. This confirms conjectures of Seed and Lambert-Cole. Along the way we prove two other conjectures about the structure of Szab{ó} homology.
Colored knot Floer homology: structures and examples
本文利用无限全扭转下 $(n,mn)$-缆线链环 Floer 同调的余极限定义 $n$ 色纽结 Floer 同调，证明其模结构，并描述 L-空间纽结情形及交叉变换映射。
Heegaard-Floer homology and string links
我们将纽结弗洛尔同调推广到D^{2} \times I中的弦链和任意三个流形中的d-基链，而不需要对分支的零同调作任何假设.对于纽结弗洛尔同调，我们得到了用弦环的挠率来描述所得到的同调群（在D^{2} \times I中）的欧拉示性数。此外，状态求和的方法被描述为使用等效的考夫曼状态。此外，我们研究了辫子的situation，证明了交替字符串链接的欧拉示性数决定的同源性，并开发类似的组成公式和长的准确序列，在结弗洛尔同调。
Knots which admit a surgery with simple knot Floer homology groups
本文证明了，如果对具有Seifert亏格g（K）的同调球面X中的纽结K进行正积分运算，得到X_n（K）=Y中的诱导纽结K_n具有简单的弗洛尔同调，则n应≥ 2 g（K）.此外，若X是标准球面，则三元流形Y是L-空间，K的Heegaard弗洛尔同调群由其Alexander多项式决定.
In a previous paper, Vértesi and the first author used grid-like Heegaard diagrams to define tangle Floer homology, which associates to a tangle $T$ a differential graded bimodule $\widetilde{\mathrm{CT}} (T)$. If $L$ is obtained by gluing together $T_1, \dotsc, T_m$, then the knot Floer homology $\hat{\mathrm{HFK}}(L)$ of $L$ can be recovered from $\widetilde{\mathrm{CT}} (T_1), \dotsc, \widetilde{\mathrm{CT}} (T_m)$. In the present paper, we prove combinatorially that tangle Floer homology satisfies unoriented and oriented skein relations, generalizing the skein exact triangles for knot Floe…
Naturality and functoriality in involutive Heegaard Floer homology
本文证明对合赫戈弗洛尔同调的一阶自然性，构造配边诱导的良定义映射，并将自然性与函子性结果推广到纽结和链环的对合弗洛尔理论。
The aim of this paper is to study the behavior of knot Floer homology under Murasugi sum. We establish a graded version of Ni's isomorphism between the extremal knot Floer homology of Murasugi sum of two links and the tensor product of the extremal knot Floer homology groups of the two summands. We further prove that $τ=g$ for each summand if and only if $τ=g$ holds for the Murasugi sum (with $τ$ and $g$ defined appropriately for multi-component links). Some applications are presented.
Using bordered Floer theory, we give a combinatorial construction and proof of invariance for the hat version of Heegaard Floer homology. As a part of the proof, we also establish combinatorially the invariance of the linear-categorical representation of the strongly-based mapping class groupoid given by the same theory.
A combinatorial description of the $U^2=0$ version of Heegaard Floer homology
我们表明，每个3-流形承认Heegaard图中的截断版本的Heegaard弗洛尔同调（当全息磁盘通过基点最多一次）可以计算组合。
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