Instanton homology and the Alexander polynomial
Yuhan Lim
我们证明了由Kronheimer和Mrowka定义的瞬子结同调KHI（K）（结，缝合和切除，预印本），恢复了3-球面中结K的Alexander多项式。
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.
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Using Freedman and Quinn's result for $\mathbb{Z}$-homology 3-spheres, we show that a 3-dimensional homology handle with trivial Alexander polynomial bounds a homology $S^1\times D^3$. As a consequence, a distinguished homology handle with trivial Alexander polynomial is topologically null $\widetilde{H}$-cobordant.
This paper shows how the Formal Knot Theory state model for the Alexander-Conway polynomial is related to Knot Floer Homology. In particular we prove a parity result about the states in this model that clarifies certain relationships of the model with Knot Floer Homology.
Instanton Floer homology and the Alexander polynomial
三球纽结的瞬子弗洛尔同调是一个模2正则分次的向量空间。它带有一个著名的自同态的甚至程度，产生于2维同调类所代表的塞弗特表面。弗洛尔同调分解为这个自同态的广义本征空间的直和。我们证明了这些广义本征空间的欧拉示性数是纽结的Alexander多项式的系数。在其他应用中，我们推断，瞬子同源检测纤维结。
In the early 2000's Cochran and Harvey introduced non-commutative Alexander polynomials for 3-manifolds. Their degrees give strong lower bounds on the Thurston norm. In this paper we make the case that the vanishing of a certain Novikov-Sikorav homology module is the correct notion of a monic non-commutative Alexander polynomial. Furthermore we will use the opportunity to give new proofs of several statements about Novikov-Sikorav homology in the three-dimensional context.
The Magnus representation and higher-order Alexander invariants for homology cobordisms of surfaces
从曲面到自身的带有边界标记的同调配边集具有自然幺半群结构。为了研究这个幺半群的结构，我们定义并研究了它的Magnus表示和Reidemeister扭转不变量，通过推广Kirk-Livingston-Wang关于弦链的Gassner表示的论点。此外，通过应用Cochran和Harvey的高阶（非交换）亚历山大不变量的框架，我们提取了几条信息的幺半群和相关对象。
Twisted Alexander Polynomials, Ribbon Homology Cobordisms, and the Thurston Norm
本文利用扭转亚历山大多项式证明，若存在从Y_-到Y_+的绳索同调柄结构且Y_-不可约，则Y_-的图松规范单位球包含Y_+的。同时证明Y_+的纤维类对应于Y_-的。
Polynomial cocycles of Alexander quandles and applications
余圈是由亚历山大量子点的多项式表达式构造的。作为应用，证明了某些quandle同调群的非平凡性，并研究了纽结的quandle上循环不变量。特别地，对于无限Quandle族，证明了Quandle同调群的非平凡性。
Mutation and instanton knot homology
我们表明，瞬子结的同源性是突变不变的，作为第三作者的早期工作的结果。
Nilpotency in instanton homology, and the framed instanton homology of a surface times a circle
本文在 Muñoz 工作的基础上计算了瞬子 Floer 同调中自同态 $u^2-64$ 的幂零次数，并进一步计算了非平凡丛情形下曲面与圆周乘积的带框架瞬子同调，揭示了其与 $u^2-64$ 核的紧密联系。
Cobordism maps in Khovanov homology and singular instanton homology I
本文定义了瞬子立方复形上的柄映射作为滤过链映射，证明其在Khovanov同调和奇异瞬子理论中恢复了柄映射，并计划在后续研究中扩展至浸没柄。
Knot homology groups from instantons
对于SU（N）的每一个部分旗流形，利用具有余维2奇点的瞬子，我们定义了三维流形中纽结的弗洛尔同调理论.
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。
Heegaard Floer homology of (n,n)-torus links: computations and questions
本文研究了$(n, n)$ -torus链环的Heegaard Floer链环同调.在$\mathbb{R}^{n}$中，支持非平凡同调的Alexander多重分次构成了一串$n-1$单位超立方体，我们计算了几乎所有Alexander分次中同调的秩和分次.我们还猜想一个完整的描述链接同源性，并提供了一些支持这一猜想。本文摘自作者2007年的博士论文。论文，并包含几个开放的问题。
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。
Two-fold quasi-alternating links, Khovanov homology and instanton homology
本文引入一类严格包含拟交错链环的新链环类，证明其模2约化Khovanov同调总是薄的，并计算其双重分支覆盖的带框架瞬子同调，同时给出存在非平凡实3平面丛时的计算及谱序列证据。
Novikov homology, twisted Alexander polynomials and Thurston cones
我们继续研究扭曲的诺维科夫同调，介绍了在我们的联合文件与H.戈达（arXiv：数学。DG/0312374），及其推广。发达的代数技术的主要应用是3流形的拓扑结构。特别地，我们证明了欧拉示性数为零的三维流形M的扭Novikov同调为零的充要条件是M的基本群的扭Alexander多项式为幺半群.讨论了三维流形的1-上同调的Thurston范数与该流形的扭曲Novikov同调之间的关系。
On Alexander modules and Blanchfield forms of null-homologous knots in rational homology spheres
本文给出了有理同调球面上零同调纽结的Alexander模的一个分类。我们刻画了具有Blanchfield形式$ϕ$的模A，以及使得存在唯一的$(A,ϕ)$同构类的模A，并证明了对于其它模A，存在无穷多个这样的类.所有这些$(A,ϕ)$都是通过有理同调球面上的显式纽结来实现的。
In this paper, we introduce the annular instanton Floer homology which is defined for links in a thickened annulus. It is an analogue of the annular Khovanov homology. A spectral sequence whose second page is the annular Khovanov homology and which converges to the annular instanton Floer homology is constructed. As an application of this spectral sequence, we prove that the annular Khovanov homology detects the unlink in the thickened annulus (assuming all the components are null-homologous). Another application is a new proof of Grigsby and Ni's result that tangle Khovanov homology distingui…
We proved that the dimension of the F-vector space J#(G) for a plane trivalent graph G, defined by Kronheimer and Mrowka using their SO(3) instanton Floer homology, is equal to the number of Tait colorings of G.
On instanton homology of corks W_n
我们考虑一族软木塞，记为$W_n$，由Akbulut和Yasui构造。每个软木塞通过其边界$Σ_n = \partial W_n$上的扭曲$τ$在光滑4-流形上产生一个奇异结构。计算了$Σ_n$的瞬子弗洛尔同调，证明了由$τ: Σ_n \rightarrow Σ_n$在瞬子弗洛尔同调上诱导的映射是非平凡的.
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.
In this article, we express the Alexander polynomial of null-homologous long knots in punctured rational homology $3$-spheres in terms of integrals over configuration spaces. To get such an expression, we use a previously established formula, which gives generalized Bott-Cattaneo-Rossi invariants in terms of the Alexander polynomial and vice versa, and we relate these Bott-Cattaneo-Rossi invariants to the perturbative expansion of Chern-Simons theory.
Suppose H is an admissible Heegaard diagram for a balanced sutured manifold (M,γ). We prove that the number of generators of the associated sutured Heegaard Floer complex is an upper bound on the dimension of the sutured instanton homology of (M,γ). It follows, in particular, that strong L-spaces are instanton L-spaces.
We prove an excision theorem for the singular instanton Floer homology that allows the excision surfaces to intersect the singular locus. This is an extension of the non-singular excision theorem by Kronheimer and Mrowka and the genus-zero singular excision theorem by Street. We use the singular excision theorem to define an instanton Floer homology theory for sutured manifolds with tangles. As applications, we prove that the annular Khovanov homology (1) detects the unlink, (2) detects the closure of the trivial braid, and (3) distinguishes braid closures from other links; we also prove that …
Ring structures in singular instanton homology
本文计算闭定向曲面上奇异瞬子 Floer 同调的环结构，并据此证明 n=1 时的切除公式，解决该理论切除公式的最后未知情形。
Filtrations on instanton homology
在作者的早期工作中，一个结或链接的Khovanov复体出现在邻接瞬子同调的谱序列的第一页。Khovanov同源的量子和（co）同调分级不生存的分级，但我们表明，他们生存的过滤。
On the knot Floer homology of a class of satellite knots
纽结弗洛尔同调是三球面纽结的不变量，其中欧拉示性数是纽结的Alexander-Conway多项式。本文的目的是研究一类卫星纽结的这种同调，从而了解卫星、伴点和模式的Alexander-Conway多项式之间的某种关系是如何在纽结的弗洛尔同调的水平上推广的.我们还使用我们的意见，研究一个经典的几何不变，塞弗特属，我们的卫星结。
Chern-Simons invariants, SO(3) instantons, and Z/2-homology cobordism
我们回顾了Fintushel和Stern的SO（3）瞬子规范理论，并将其转换到具有圆柱端的4-流形的背景下。给出了Z/2-同调3-球面的Z/2-同调配边群的应用.
The Alexander trick for homology spheres
本文证明在维度至少为6时，固定边界的紧致可缩流形同胚群可缩，并由此推导出单侧$h$-配边的强唯一性。
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 3-manifolds includes all Seifert fibered rational homology spheres with base orbifold $S^2$ (we establish the isomorphism for the remaining Seifert fibered rational homology spheres$\unicode{x2014}$with base $\mathbb{RP}^2$$\unicode{x2014}$directly). Our proof utilizes lattice homology, and relies on a decomposition theorem for instanton Floer cobordism maps recen…
A spectral sequence is established, whose $E_{2}$ page is Bar-Natan's variant of Khovanov homology and which abuts to a deformation of instanton homology for knots and links. This spectral sequence arises as a specialization of a spectral sequence whose $E_{2}$ page is a characteristic-2 version of $F_{5}$ homology, in Khovanov's classification.
Given a transverse knot $K$ in a three dimensional contact manifold $(Y,α)$, in [13] Colin, Ghiggini, Honda and Hutchings define a hat version of embedded contact homology for $K$, that we call $\widehat{ECK}(K,Y,α)$, and conjecture that it is isomorphic to the knot Floer homology $\widehat{HFK}(K,Y)$. We define here a full version $ECK(K,Y,α)$ and generalise the definitions to the case of links. We prove then that, if $Y = S^3$, $ECK$ and $\widehat{ECK}$ categorify the (multivariable) Alexander polynomial of knots and links, obtaining expressions analogue to that for knot and link Floer homol…
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…
We compute different versions of link Floer homology $HFL^{-}$ and $\widehat{HFL}$ for any $L$-space link with two components. The main approach is to compute the $h$-function of the filtered chain complex which is determined by the Alexander polynomials of every sublink of the $L$-space link. As an application, Thurston polytope and Thurston norm of any 2-component $L$-space link are explicitly determined by Alexander polynomials of the link and the link components.
Framed instanton homology of surgeries on L-space knots
本文计算了具有瞬子L-空间手术的纽结在所有整数手术上的带标架瞬子Floer同调，并证明若纽结同时具有Heegaard Floer和瞬子Floer的L-空间手术，则两种理论在所有整数手术上一致。
We define two concordance invariants of knots using framed instanton homology. These invariants $ν^\sharp$ and $τ^\sharp$ provide bounds on slice genus and maximum self-linking number, and the latter is a concordance homomorphism which agrees in all known cases with the $τ$ invariant in Heegaard Floer homology. We use $ν^\sharp$ and $τ^\sharp$ to compute the framed instanton homology of all nonzero rational Dehn surgeries on: 20 of the 35 nontrivial prime knots through 8 crossings, infinite families of twist and pretzel knots, and instanton L-space knots; and of 19 of the first 20 closed hyper…
Knot cobordism, torsion order and framed instanton homology
本文为瞬子纽结同调构造配边映射，恢复挠阶不等式，并据此计算交错纽结Dehn手术的带框瞬子Floer同调。
Surgery Exact Triangles in Instanton Theory
本文证明了联系纽结instanton Floer同调与手术instanton同调的正合三角形，这是整数系数下的首个此类结果，并据此导出Poincaré同调球在$\mathbb{Z}/2$-系数下非instanton $L$-空间。
Cobordism maps in Khovanov homology and singular instanton homology II
本文将嵌入配边映射推广到浸入配边，证明其与奇异瞬子同调相容，并据此证明二桥环面结concordance诱导的Khovanov同调映射单射，以及相对奇异曲面在正扭转移动下的奇异性保持。
Floer homology for 2-torsion instanton invariants
我们构造了一个变体的弗洛尔同调群，并证明了一个胶合公式的变体的唐纳森不变量。作为推论，对于满足Donaldson不变量条件的4-流形的连通和，Donaldson不变量的变体是非平凡的。我们还证明了具有某些同调3-球面和某些相交形式的紧自旋4-流形的不存在性。
Asaeda, Przytycki and Sikora defined a generalization of Khovanov homology for links in $I$-bundles over compact surfaces. We prove that for a link $L\subset (-1,1)\times T^2$, the Asaeda-Przytycki-Sikora homology of $L$ has rank $2$ with $\mathbb{Z}/2$-coefficients if and only if $L$ is isotopic to an embedded knot in $\{0\}\times T^2$.
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 …
Knot concordance, the point class in instanton homology and Donaldson invariants
本文通过Donaldson不变量与Floer瞬子同调构造三维球面中纽结的不变量$φ$，证明其基本性质，并说明它与Froyshov不变量的一个特殊情形相同。
We will prove that, for a $2$ or $3$ component $L$-space link, $HFL^-$ is completely determined by the multi-variable Alexander polynomial of all the sub-links of $L$, as well as the pairwise linking numbers of all the components of $L$. We will also give some restrictions on the multi-variable Alexander polynomial of an $L$-space link. Finally, we use the methods in this paper to prove a conjecture by Yajing Liu classifying all $2$-bridge $L$-space links.
Instantons and Khovanov homology in $\mathbb{RP}^3$
本文研究 $\mathbb{RP}^3$ 中链环的瞬子 Floer 同调，计算 Kronheimer–Mrowka 谱序列第二页，并据此证明 Khovanov 同调可检测平凡纽结与射影平凡纽结。
Instantons and rational homology spheres
论文研究了有理同调球面的equivariant instanton homology groups的不变性，并证明其在3流形cobordism下的泛函性，同时定义了新的瞬子不变量。
Instanton Floer homology for lens spaces
我们构造了透镜空间$L(p,q)$的瞬子弗洛尔同调.作为应用，我们证明了$X = \CP^2 # \CP^2$不允许分解$X = X_1 \cup X_2$ .这里$X_1$和$X_2$是定向的、单连通的、非自旋的4-流形，具有$b^+ = 1$和边界$L(p, 2)$，并且$p$是$16N+1$形式的素数。
A spectral sequence is established, from Bar-Natan's variant of Khovanov homology to a deformation of instanton homology for knots and links. This spectral sequence arises as a specialization of a spectral sequence from a characteristic-2 version of $F_5$ homology, in Khovanov'sclassification. Concordance invariants of knots are derived from the corresponding instanton homology groups, including a 1-parameter family of homomorphisms $f_r$ from the concordance group to the reals, having the potential to provide independent bounds on the genus and number of double points for immersed surfaces wi…
The equivariant rho-invariants studied in this paper are a version of the classical rho-invariants of Atiyah, Patodi, and Singer in the presence of an isometric involution. We compute these rho-invariants for all involutions on the 3-dimensional lens spaces with 1-dimensional fixed point sets, as well as for some involutions on Brieskorn homology spheres. As an application, we compute the generators and Floer gradings in the singular instanton chain complex of (p,q)-torus knots with odd p and q.
Filtered instanton homology and cosmetic surgery
该研究利用带Chern-Simons滤过的瞬子Floer同调排除了美容手术猜想中±1/n手术的可能，将猜想简化为亏格2平凡Alexander多项式纽结的±2手术情形，同时定义了新的三维流形不变量与瞬子同调手术关系，并推广到S²×S¹中的纽结。
Relations in singular instanton homology
本文计算了 $S^1 \ imes S^2$ 中奇数 n-辫的带局部系数奇异瞬子同调，用代数曲线坐标环描述其结构，并预期等价于某 Fano 簇的量子上同调环。
We prove that for "most" closed 3-dimensional manifolds $M$, the existence of a closed non singular one-form in a given cohomology class $u\in H^1 (M,\bf R)$ is equivalent to the fact that every twisted Alexander polynomial $Δ^H(M,u) \in {\bf Z}[G/\ker u]$ associated to a normal subgroup with finite index $H < π_1(M)$ has a unitary $u$-minimal term.
We show that bordered Floer homology provides a categorification of a TQFT described by Donaldson. This, in turn, leads to a proof that both the Alexander module of a knot and the Seifert form are completely determined by Heegaard Floer theory.
Equivariant aspects of singular instanton Floer homology
本文利用$SU(2)$奇异瞬子规范理论为整数同调3-球面中的纽结构造等变不变量，包括类似纽结Floer同调的三元群和Frøyshov型配边不变量，并借助ADHM描述刻画球面纽结的模空间。
On elementary invariants of genus one knots and Seifert surfaces
本文提出了一种用于研究一阶结和塞弗特面的亚历山大形式不变量，通过分析三维同调柄体的性质来证明其不变性。
On Heegaard Floer homology and Seifert fibered surgeries
我们探讨了某些限制结在三个领域承认非平凡的塞弗特纤维手术。这些限制源于塞弗特纤维空间的Heegaard弗洛尔同调，因此它们对这种纽结的Alexander多项式和纽结的弗洛尔同调都有影响。特别是，我们表明，某些多项式从来没有亚历山大多项式的结承认同源三球塞弗特纤维手术。另一方面，纽结弗洛尔同调限制也适用于亚历山大多项式没有给出信息的情况，例如木下寺坂纽结。
Given an oriented link in the 3-sphere, the Euler characteristic of its link Floer homology is known to coincide with its multivariate Alexander polynomial, an invariant only defined up to a sign and powers of the variables. In this paper, we get rid of this ambiguity by proving that this Euler characteristic is equal to the so-called Conway function, the representative of the multivariate Alexander polynomial introduced by Conway in 1970 and explicitly constructed by Hartley in 1983. This is achieved by creating a model of the Conway function adapted to rectangular diagrams, which is then com…
We construct a bigraded spectral sequence from the gl(0)-homology to knot Floer homology. This spectral sequence is of Bockstein type and comes from a subtle manipulation of coefficients. The main tools are quantum traces of foams and of singular Soergel bimodules.
Framed instanton homology and Frøyshov's invariant
本文研究了结的Dehn手术中框架瞬子同调与Froyshov不变量的关系，并证明了非退化$SU(2)$-abelian的限制条件。
Instantons, indefinite 4-manifolds, and Dehn surgery
该研究利用基于模2瞬子同调的Froyshov不变量$q_3$，证明存在Dehn手术数可任意大的双曲整数同调球，并将相关不等式扩展至有理同调球。
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