Knots, sutures and excision

带边缝合流形的guts的把柄数与近纤维纽结

Handle numbers of guts of sutured manifolds and nearly fibered knots

本研究将Haken定理推广到缝合流形Heegaard分解的乘积环面与圆盘，证明不可约缝合流形把柄数等于其guts的把柄数，分类了S³中带环面边界的约化缝合流形，并揭示近纤维纽结的把柄数与不可压缩Seifert曲面的多样性。

On sutured Floer homology and the equivalence of Seifert surfaces

We study the sutured Floer homology invariants of the sutured manifold obtained by cutting a knot complement along a Seifert surface, R. We show that these invariants are finer than the "top term" of the knot Floer homology, which they contain. In particular, we use sutured Floer homology to distinguish two non-isotopic minimal genus Seifert surfaces for the knot 8_3. A key ingredient for this technique is finding appropriate Heegaard diagrams for the sutured manifold associated to the complement of a Seifert surface.

Decomposing sutured monopole and Instanton Floer homologies

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 …

Homological actions on sutured Floer homology

We define the action of the homology group $H_1(M,\partial M)$ on the sutured Floer homology $SFH(M,γ)$. It turns out that the contact invariant $EH(M,γ,ξ)$ is usually sent to zero by this action. This fact allows us to refine an earlier result proved by Ghiggini and the author. As a corollary, we classify knots in $#^n(S^1\times S^2)$ which have simple knot Floer homology groups: They are essentially the Borromean knots.

Knot homologies in monopole and instanton theories via sutures

In this paper we construct possible candidates for the minus versions of monopole and instanton knot Floer homologies. For a null-homologous knot $K\subset Y$ and a base point $p\in K$, we can associate the minus versions, $\underline{\rm KHM}^-(Y,K,p)$ and $\underline{\rm KHI}^-(Y,K,p)$, to the triple $(Y,K,p)$. We prove that a Seifert surface of $K$ induces a $\mathbb{Z}$-grading, and there is an $U$-map on the minus versions, which is of degree $-1$. We also prove other basic properties of them. If $K\subset Y$ is not null-homologous but represents a torsion class, then we can also construc…

缆链、环带与缝合 Floer 同调

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。

Sutured Floer homology distinguishes between Seifert surfaces

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 answers a question of Juhász. More precisely, we show that the Euler characteristic of the sutured Floer homology of the complementary manifolds distinguishes between the two surfaces, as does the sutured Floer polytope introduced by Juhász. Actually, we exhibit an infinite family of knots with pairs of Seifert surfaces that can be distinguished by the Euler cha…

关于 sutured 流形中的 Heegaard Floer 极小纽结

On Heegaard Floer minimal knots in sutured manifolds

本文在 Heegaard Floer 同调框架下分类了平衡 sutured 流形中的极小纽结，与 instanton Floer 分类一致，支持两种同调同构的猜想，并研究了 $S^1\ imes S^2$ 中的链接 Floer 同调植物学问题。

Instanton homology and the Alexander polynomial

We prove that the instanton knot homology KHI(K) as defined by Kronheimer and Mrowka (Knots, sutures and excision, preprint), recovers the Alexander polynomial for knots K in the 3-sphere.

Seifert surfaces distinguished by sutured Floer homology but not its Euler characteristic

In this paper we find a family of knots with trivial Alexander polynomial, and construct two non-isotopic Seifert surfaces for each member in our family. In order to distinguish the surfaces we study the sutured Floer homology invariants of the sutured manifolds obtained by cutting the knot complements along the Seifert surfaces. Our examples provide the first use of sutured Floer homology, and not merely its Euler characteristic(a classical torsion), to distinguish Seifert surfaces. Our technique uses a version of Floer homology, called Longitude Floer Homology in a way that enables us to byp…

Homology cylinders and sutured manifolds for homologically fibered knots

Sutured manifolds defined by Gabai are useful in the geometrical study of knots and 3-dimensional manifolds. On the other hand, homology cylinders are in an important position in the recent theory of homology cobordisms of surfaces and finite-type invariants. We study a relationship between them by focusing on sutured manifolds associated with a special class of knots which we call {\it homologically fibered knots}. Then we use invariants of homology cylinders to give applications to knot theory such as fibering obstructions, Reidemeister torsions and handle numbers of homologically fibered kn…

缝合瞬子同调上的同调作用

A homological action on sutured instanton homology

本文在缝合瞬子Floer同调上定义了良定义的同调作用，揭示其在连通和与流形分解下的良好性质，并证明瞬子纽结同调可检测二分量链环的分裂。

Instanton Floer homology, sutures, and Heegaard diagrams

This paper establishes a new technique that enables us to access some fundamental structural properties of instanton Floer homology. As an application, we establish, for the first time, a relation between the instanton Floer homology of a $3$-manifold or a null-homologous knot inside a $3$-manifold and the Heegaard diagram of that $3$-manifold or knot. We further use this relation to compute the instanton knot homology of some families of $(1,1)$-knots, including all torus knots in $S^3$, which were mostly unknown before. As a second application, we also study the relation between the instanto…

Sutured Floer homology and invariants of Legendrian and transverse knots

Using contact-geometric techniques and sutured Floer homology, we present an alternate formulation of the minus and plus version of knot Floer homology. We further show how natural constructions in the realm of contact geometry give rise to much of the formal structure relating the various versions of Heegaard Floer homology. In addition, to a Legendrian or transverse knot K in a contact manifold (Y,ξ), we associate distinguished classes EHL(K) in the minus-version of knot floer homology and EHIL(K) in the plus version, which are each invariant under Legendrian or transverse isotopies of K. Th…

The sutured Floer homology polytope

In this paper, we extend the theory of sutured Floer homology developed by the author. We first prove an adjunction inequality, and then define a polytope P(M,g) in H^2(M,\partial M; R) that is spanned by the Spin^c-structures which support non-zero Floer homology groups. If (M,g) --> (M',g') is a taut surface decomposition, then a natural map projects P(M',g') onto a face of P(M,g); moreover, if H_2(M) = 0, then every face of P(M,g) can be obtained in this way for some surface decomposition. We show that if (M,g) is reduced, horizontally prime, and H_2(M) = 0, then P(M,g) is maximal dimension…

Earrings, sutures and pointed links

We prove a rank inequality on the instanton knot homology and the Khovanov homology of a link in $S^3$. The key step of the proof is to construct a spectral sequence relating Baldwin-Levine-Sarkar's pointed Khovanov homology to a singular instanton invariant for pointed links.

Instanton and the depth of taut foliations

Sutured instanton Floer homology was introduced by Kronheimer and Mrowka. In this paper, we prove that for a taut balanced sutured manifold with vanishing second homology, the dimension of the sutured instanton Floer homology provides a bound on the minimal depth of all possible taut foliations on that balanced sutured manifold. The same argument can be adapted to the monopole and even the Heegaard Floer settings, which gives a partial answer to one of Juhasz's conjectures. Using the nature of instanton Floer homology, on knot complements, we can construct a taut foliation with bounded depth, …

瞬子Floer同调、缝合与欧拉示性数

Instanton Floer homology, sutures, and Euler characteristics

本文为平衡缝合流形上的Floer同调提出公理化定义，证明其欧拉示性数由公理唯一确定，并由此恢复链环的多变量Alexander多项式，同时构造有理零同调纽结的瞬子同调分次。

Concordance surgery and the Ozsváth-Szabó 4-manifold invariant

We compute the effect of concordance surgery, a generalization of knot surgery defined using a self-concordance of a knot, on the Ozsváth-Szabó 4-manifold invariant. The formula involves the graded Lefschetz number of the concordance map on knot Floer homology. The proof uses the sutured Floer TQFT, and a version of sutured Floer homology perturbed by a 2-form.

缝合瞬子同调的一个增强欧拉示性数

An enhanced Euler characteristic of sutured instanton homology

本文为缝合瞬子同调构造了基于挠元的分解，定义增强欧拉示性数并证明其等于缝合 Floer 同调的欧拉示性数，从而给出更好的维数下界，并应用于检测平凡结及验证 KHI 与 hat-HFK 同构猜想。

Guts in Sutured Decompositions and the Thurston Norm

We construct an invariant called guts for second homology classes in irreducible 3-manifolds with toral boundary and non-degenerate Thurston norm. We prove that the guts of second homology classes in each Thurston cone are invariant under a natural condition. We show that the guts of different homology classes are related by sutured decompositions. As an application, an invariant of knot complements is given and is computed in a few interesting cases. Besides, we show that the dimension of a maximal simplex in a Kakimizu Complex is an invariant.

On the Colored Jones Polynomial, Sutured Floer homology, and Knot Floer homology

Let K in S^3 be a knot, and let \widetilde{K} denote the preimage of K inside its double branched cover, Σ(K). We prove, for each integer n > 1, the existence of a spectral sequence from Khovanov's categorification of the reduced n-colored Jones polynomial of the mirror of K to the knot Floer homology of (Σ(K),\widetilde{K}) (when n odd) and to (S^3, K # K) (when n even). A corollary of our result is that Khovanov's categorification of the reduced n-colored Jones polynomial detects the unknot whenever n>1.

The decategorification of sutured Floer homology

We define a torsion invariant T for every balanced sutured manifold (M,g), and show that it agrees with the Euler characteristic of sutured Floer homology SFH. The invariant T is easily computed using Fox calculus. With the help of T, we prove that if (M,g) is complementary to a Seifert surface of an alternating knot, then SFH(M,g) is either 0 or Z in every spin^c structure. T can also be used to show that a sutured manifold is not disk decomposable, and to distinguish between Seifert surfaces. The support of SFH gives rise to a norm z on H_2(M, \partial M; R). Then T gives a lower bound on th…

Floer homology and surface decompositions

Sutured Floer homology, denoted by SFH, is an invariant of balanced sutured manifolds previously defined by the author. In this paper we give a formula that shows how this invariant changes under surface decompositions. In particular, if (M, γ)--> (M', γ') is a sutured manifold decomposition then SFH(M',γ') is a direct summand of SFH(M, γ). To prove the decomposition formula we give an algorithm that computes SFH(M,γ) from a balanced diagram defining (M,γ) that generalizes the algorithm of Sarkar and Wang. As a corollary we obtain that if (M, γ) is taut then SFH(M,γ) is non-zero. Other applica…

Comparing invariants of Legendrian knots

We prove the equivalence of the invariants EH(L) and LOSS-(L) for oriented Legendrian knots L in the 3-sphere equipped with the standard contact structure, partially extending a previous result by Stipsicz and Vertesi. In the course of the proof we relate the sutured Floer homology groups associated with a knot complement and knot Floer homology, and define intermediate Legendrian invariants.

秩三瞬子、表示与缝合

Rank three instantons, representations and sutures

本文研究了 $U(3)$ 瞬子 Floer 同调的环结构，部分解决特征值猜想，建立缝合流形的曲面分解定理，并据此证明任意整数同调球中纽结群存在到 $SU(3)$ 的特定非阿贝尔表示。

关于接触手术与纽结Floer不变量

On contact surgery and knot Floer invariants

本文建立了接触手术与Legendrian纽结Floer不变量之间的非消失关系，利用缝合Floer同调及极限构造，证明了接触不变量非零蕴含相应Legendrian不变量非零。

Naturality in sutured monopole and instanton homology

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 refinements of these invariants which assign much richer algebraic objects called projectively transitive systems of modules to balanced sutured manifolds and isomorphisms of such systems to diffeomorphisms of balanced sutured manifolds. Our work provides the foundation for extending these sutured Floer theories to other interesting functorial frameworks as well, and …

Annular Khovanov homology and knotted Schur-Weyl representations

Let L be a link in a thickened annulus. We show that its sutured annular Khovanov homology carries an action of the exterior current algebra of the Lie algebra sl_2. When L is an m-framed n-cable of a knot K in the three-sphere, its sutured annular Khovanov homology carries a commuting action of the symmetric group S_n. One therefore obtains a "knotted" Schur-Weyl representation that agrees with classical sl_2 Schur-Weyl duality when K is the Seifert-framed unknot.

Khovanov homology is an unknot-detector

We prove that a knot is the unknot if and only if its reduced Khovanov cohomology has rank 1. The proof has two steps. We show first that there is a spectral sequence beginning with the reduced Khovanov cohomology and abutting to a knot homology defined using singular instantons. We then show that the latter homology is isomorphic to the instanton Floer homology of the sutured knot complement: an invariant that is already known to detect the unknot.

瞬子弗勒尔同调的纽结手术公式 I：主定理

Knot surgery formulae for instanton Floer homology I: the main theorem

本文针对3-流形Y中满足[K]=0∈H₁(Y;ℚ)且m≠0的纽结K，证明了带边瞬子同调Iᵛ(Yₘ(K))的整手术公式，其证明基于缝合瞬子同调SHI和导出范畴的八面体引理，还得到相关正合三角形及独立价值的新结果。

A refinement of sutured Floer homology

We introduce a refinement of the Ozsvath-Szabo complex associated to a balanced sutured manifold $(X,τ)$ by Juhasz. An algebra $A_τ$ is associated to the boundary of a sutured manifold and a filtration of its generators by $H^2(X,\partial X;\Z)$ is defined. For a fixed Spin^c structure $s$ over the manifold $X'$, which is obtained from $X$ by filling out the sutures, the Ozsvath-Szabo chain complex $CF(X,τ,s)$ is then defined as a chain complex with coefficients in $A_τ$ and filtered by $\SpinC(X,τ)$. The filtered chain homotopy type of this chain complex is an invariant of $(X,τ)$ and the Spi…

Sutured annular Khovanov-Rozansky homology

We introduce an sl(n) homology theory for knots and links in the thickened annulus. To do so, we first give a fresh perspective on sutured annular Khovanov homology, showing that its definition follows naturally from trace decategorifications of enhanced sl(2) foams and categorified quantum gl(m), via classical skew Howe duality. This framework then extends to give our annular sl(n) link homology theory, which we call sutured annular Khovanov-Rozansky homology. We show that the sl(n) sutured annular Khovanov-Rozansky homology of an annular link carries an action of the Lie algebra sl(n), which…

Khovanov homology, sutured Floer homology, and annular links

Lawrence Roberts, extending the work of Ozsvath-Szabo, showed how to associate to a link, L, in the complement of a fixed unknot, B, in S^3, a spectral sequence from the Khovanov homology of a link in a thickened annulus to the knot Floer homology of the preimage of B inside the double-branched cover of L. In a previous paper, we extended Ozsvath-Szabo's spectral sequence in a different direction, constructing for each knot K in S^3 and each positive integer n, a spectral sequence from Khovanov's categorification of the reduced, n-colored Jones polynomial to the sutured Floer homology of a red…

On the equivalence of contact invariants in sutured Floer homology theories

We recently defined an invariant of contact manifolds with convex boundary in Kronheimer and Mrowka's sutured monopole Floer homology theory. Here, we prove that there is an isomorphism between sutured monopole Floer homology and sutured Heegaard Floer homology which identifies our invariant with the contact class defined by Honda, Kazez and Matić in the latter theory. One consequence is that the Legendrian invariants in knot Floer homology behave functorially with respect to Lagrangian concordance. In particular, these invariants provide computable and effective obstructions to the existence …

具有奇模式的纽结的实Seiberg-Witten Floer同伦类型的卫星公式

A satellite formula for real Seiberg-Witten Floer homotopy types

本文建立了具有奇模式的纽结的实Seiberg-Witten Floer同伦类型的卫星公式，基于实Floer同伦类型的切除定理，导出了纽结配边理论的应用，并证明配边不变量仅依赖于零框架手术。

Knot Floer homology detects genus-one fibred knots

Ozsvath and Szabo conjectured that knot Floer homology detects fibred knots. We propose a strategy to approach this conjecture based on Gabai's theory of sutured manifold decomposition and contact topology. We implement this strategy for genus-one knots, obtaining as a corollary that, if rational surgery on a knot $K$ gives the Poincare homology sphere $Σ(2,3,5)$, then $K$ is the left-handed trefoil knot.

Monopole Floer homology and Legendrian knots

We use monopole Floer homology for sutured manifolds to construct invariants of Legendrian knots in a contact 3-manifold. These invariants assign to a knot K in Y elements of the monopole knot homology KHM(-Y,K), and they strongly resemble the knot Floer homology invariants of Lisca, Ozsváth, Stipsicz, and Szabó. We prove several vanishing results, investigate their behavior under contact surgeries, and use this to construct many examples of non-loose knots in overtwisted 3-manifolds. We also show that these invariants are functorial with respect to Lagrangian concordance.

Instanton homology and knot detection on thickened surfaces

Suppose $Σ$ is a compact oriented surface (possibly with boundary) that has genus zero, and L is a link in the interior of $(-1,1)\timesΣ$. We prove that the Asaeda-Przytycki-Sikora (APS) homology of L has rank 2 if and only if L is isotopic to an embedded knot in $\{0\}\timesΣ$. As a consequence, the APS homology detects the unknot in $(-1,1)\timesΣ$. This is the first detection result for generalized Khovanov homology that is valid on an infinite family of manifolds, and it partially solves a conjecture in arxiv:2005.12863. Our proof is different from the previous detection results obtained …

Certifying the Thurston norm via SL(2, C)-twisted homology

We study when the Thurston norm is detected by twisted Alexander polynomials associated to representations of the 3-manifold group to SL(2, C). Specifically, we show that the hyperbolic torsion polynomial determines the genus for a large class of hyperbolic knots in the 3-sphere which includes all special arborescent knots and many knots whose ordinary Alexander polynomial is trivial. This theorem follows from results showing that the tautness of certain sutured manifolds can be certified by checking that they are a product from the point of view of homology with coefficients twisted by an SL(…

纽结Floer同调中的连通和与有向系统

Connected sums and directed systems in knot Floer homologies

本文证明了instanton knot Floer homology的基本性质，主要贡献包括推导其连通和公式、有向skein正合三角形，以及提出适用于所有Floer理论的自由有限生成链复形新模型。

实链环Floer同调

Real link Floer homology

本文为具有连通不动集的闭实三维流形中的强可逆和双周期链环定义了实链环Floer同调，推广了实Heegaard Floer同调和实带边Heegaard Floer同调，并通过实网格图给出了S^3中的组合描述，用于研究理论的结构性质及强可逆纽结的性质。

Transverse invariants and exotic surfaces in the 4-ball

Using 1-twist rim surgery, we construct infinitely many smoothly embedded, orientable surfaces in the 4-ball bounding a knot in the 3-sphere that are pairwise topologically isotopic, but not ambient diffeomorphic. We distinguish the surfaces using the maps they induce on perturbed sutured Floer homology. Along the way, we show that the cobordism map induced by an ascending surface in a Weinstein cobordism preserves the transverse invariant in knot Floer homology.

Khovanov homology detects the trefoils

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 triangle in sutured instanton homology, proven here; and Kronheimer and Mrowka's spectral sequence relating Khovanov homology with singular instanton knot homology. As a byproduct, we also strengthen a result of Kronheimer and Mrowka on $SU(2)$ representations of the knot group.

Singular crossings and Ozsváth-Szabó's Kauffman-states functor

Recently, Ozsváth and Szabó introduced some algebraic constructions computing knot Floer homology in the spirit of bordered Floer homology, including a family of algebras B(n) and, for a generator of the braid group on n strands, a certain type of bimodule over B(n). We define analogous bimodules for singular crossings. Our bimodules are motivated by counting holomorphic disks in a bordered sutured version of a Heegaard diagram considered previously by Ozsváth, Stipsicz, and Szabó.

Polytopes, dualities, and Floer homology

This article is an exposition of a body of existing results, together with an announcement of recent results. We discuss a theory of polytopes associated to bipartite graphs and trinities, developed by Kálmán, Postnikov and others. This theory exhibits a variety of interesting duality and triality relations, and extends into knot theory, 3-manifold topology and Floer homology. In recent joint work with Kálmán, we extend this story into contact topology and contact invariants in sutured Floer homology.

Knot Floer 同调、Burau 表示与量子 $\mathfrak{gl}(1 \vert 1)$

Knot Floer Homology, the Burau Representation, and Quantum $\mathfrak{gl}(1 \vert 1)$

本文通过 Fox 微积分建立 Burau 表示与 Knot Floer 同调的明确联系，定义范畴化行列式的不变量，并利用量子 $\mathfrak{gl}(1 \vert 1)$ 证明其恢复辫表示。

瞬子2- torsion与纤维化结

Instanton 2-torsion and fibered knots

研究证明了纤维化结的瞬子2-torsion性质，并通过缝合瞬子理论得到相关同调的公式，同时揭示了Alexander多项式与瞬子同调的关系及对Heegaard Floer理论的联系。

Abelian quotients of monoids of homology cylinders

A homology cylinder over a surface consists of a homology cobordism between two copies of the surface and markings of its boundary. The set of isomorphism classes of homology cylinders over a fixed surface has a natural monoid structure and it is known that this monoid can be seen as an enlargement of the mapping class group of the surface. We now focus on abelian quotients of this monoid. We show that both the monoid of all homology cylinders and that of irreducible homology cylinders are not finitely generated and moreover they have big abelian quotients. These properties contrast with the f…

亏格为一的近纤维链环

Nearly fibered links with genus one

本文分类了三维球面中满足特定欧拉示性数条件的近纤维 n 分量链环，其最大 Alexander 分次秩为二，证明受 Baldwin 和 Sivek 启发，并利用缝合 Floer 同调技术，同时计算了这些链环的链环 Floer 同调群。