Two detection results of Khovanov homology on links

Khovanov homology and diagonalisable Frobenius algebras

We give a short elementary proof that a Khovanov-type link homology constructed from a diagonalisable Frobenius algebra is degenerate.

关于Khovanov-Rozansky $sl_2$ -同调和普通Khovanov同调的注记

A note on Khovanov-Rozansky $sl_2$-homology and ordinary Khovanov homology

本文给出了Khovanov-Rozansky $sl_2$ -同调与普通Khovanov同调之间的显式同构。这个结果最初在Khovanov和Rozansky的论文\cite{KRI}中陈述，尽管细节还没有出现在文献中。主要缺失的细节是在识别$sl_2$ -同源性中的变量时提供一致的符号选择。随着$sl_2$ -同调中符号和局部取向的行为，当我们试图将它们的定义扩展到虚链时，这两个理论的行为是不同的，这似乎表明$sl_2$ -同调可能对应于霍瓦诺夫同调的不同变体。在本文中，我们描述了这两种理论，并证明它们实际上是同构的，表明可以作出连贯的选择的迹象。在这样做时，我们强调的解释$sl_2$ -复杂的作为一个立方体的决议。

On Sutured Khovanov Homology and Axis-Preserving Mutations

This paper establishes that sutured annular Khovanov homology is not invariant for braid closures under axis-preserving mutations. This follows from an explicit relationship between sutured annular Khovanov homology and the classical Burau representation for braid closures.

Instantons and Annular Khovanov Homology

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…

Szabó谱序列从奇Khovanov同调出发的积分提升

An integral lift, starting in odd Khovanov homology, of Szabó's spectral sequence

Ozsváth，Rasmussen和Szabó构造了奇Khovanov同调。它是一个链接不变量，具有相同的减少模2（甚至）Khovanov同调。Szabó从模2 Khovanov同调到另一个链同调引入了模2系数的谱序列。他从一个带过滤的链复形中得到了他的谱序列。给出了Szabó复形的一个积分提升，它提供了从奇Khovanov同调到链同调的一个谱序列，由此利用泛系数定理得到Szabó链同调. Szabó独立建造了这样的电梯（未出版）。

Khovanov同调中挠元的一个模式

A pattern for torsion in Khovanov homology

本文证明了特定增强状态之和可在Khovanov同调中产生二阶挠元，揭示了该同调理论中挠元的一种构造模式。

A spectral sequence from Khovanov homology to knot Floer homology

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.

Chromatic homology, Khovanov homology, and torsion

In the first few homological gradings, there is an isomorphism between the Khovanov homology of a link and the categorification of the chromatic polynomial of a graph related to the link. In this article, we show that the categorification of the chromatic polynomial only contains torsion of order two, and hence Khovanov homology only contains torsion of order two in the gradings where the isomorphism is defined. We also prove that odd Khovanov homology is torsion-free in its first few homological gradings.

An Introduction to Khovanov Homology

This paper is an introduction to Khovanov homology, starting with the Kauffman bracket state summation, emphasizing the Bar-Natan Canopoloy and tangle cobordism approach. The paper discusses a simplicial approach to Khovanov homology and a quantum model for it so that the graded Euler characteristic that produces the Jones polynomial from Khovanov homology becomes the trace of a unitary transformation on a Hilbert space associated with the Khovanov Homology.

任意系数虚连杆的Khovanov同调

Khovanov homology for virtual links with arbitrary coefficients

利用扭曲系数方法，我们构造了任意系数虚链路的Khovanov同调论。该方法也适用于构建Viro意义上的"非定向虚拟结“的Khovanov同源性，特别是${\bf R}P^{3}$中的结。

On the four-term relation on Khovanov homology

The goal of this paper is to prove a categorified analogue of Kontsevich's $4T$ relation on Vassiliev derivatives of Khovanov homology.

Crossing change on Khovanov homology and a categorified Vassiliev skein relation

Khovanov homology is a categorification of the Jones polynomial, so it may be seen as a kind of quantum invariant of knots and links. Although polynomial quantum invariants are deeply involved with Vassiliev (aka. finite type) invariants, the relation remains unclear in case of Khovanov homology. Aiming at it, in this paper, we discuss a categorified version of Vassiliev skein relation on Khovanov homology. More precisely, we will show that the "genus-one" operation gives rise to a crossing change on Khovanov complexes. Invariance under Reidemeister moves turns out, and it enables us to extend…

（p，-p，q）椒盐卷饼结的Khovanov同调

The Khovanov homology of (p,-p,q) pretzel knots

在本文中，我们计算了\Q上（p，-p，q）椒盐卷饼结的Khovanov同调，其中p为奇数，q为任意大.我们提供了一个猜想的Khovanov同调的一般形式的（p，-p，q）椒盐卷饼结。这些计算表明，这些结具有薄Khovanov同源性（超过\Q和\Z）。因为格林已经证明了这些纽结不是拟交错的，这就提供了一个无限类的非拟交错纽结与薄Khovanov同调。

全扭Khovanov同调中的一个符号分配

A sign assignment in totally twisted Khovanov homology

我们提升的特征-2全扭曲Khovanov同调的Roberts和Jaeger的理论与整数系数。其结果是一个复杂的计算约化奇Khovanov同调的纽结。这个复形等价于生成树复形，其微分是显式模的符号模糊性来自于需要在奇Khovanov同调的定义中选择符号分配。

确定状态圈Khovanov同调类非平凡性的图解法

Graphical methods establishing nontriviality of state cycle Khovanov homology classes

我们确定当某些状态循环表示非平凡的Khovanov同源类通过分析状态图的功能。使用这种方法，我们能够产生任意多个对角线包含非平凡状态循环同调类的双曲结。这给了下界的Khovanov宽度的结的复杂性排除计算的充分的同源性。

On the Khovanov Homology of Surgeries

We show a spectral sequence for the rational Khovanov homology of an oriented link in terms of the rational Khovanov complexes and homologies of the link surgeries along an admissible cut. As a non trivial corollary, we give an explicit splitting formula for the Jones polynomial.

Mutation-invariance of Khovanov-Floer theories

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.

Khovanov-Rozansky $\mathfrak{sl}_N$-homology for periodic links

For an $m$-periodic link $L$, we show that the Khovanov-Rozansky $\mathfrak{sl}_N$-homology carries an action of the group $\mathbb{Z}_m$. As an example of applications, we prove an analog of the periodicity criterion of Borodzik--Politarczyk using $\mathfrak{sl}_N$-homology instead of Khovanov homology.

Equivariant Khovanov Homology of Periodic Links

The purpose of this paper is to construct and study equivariant Khovanov homology - a version of Khovanov homology theory for periodic links. Since our construction works regardless of the characteristic of the coefficient ring it generalizes a previous construction by Chbili. We establish invariance under equivariant isotopies of links and study algebraic properties of integral and rational version of the homology theory. Moreover, we construct a skein spectral sequence converging to equivariant Khovanov homology and use this spectral sequence to compute, as an example, equivariant Khovanov h…

Khovanov同调的最大次数在扭曲下的性质

The behavior of the maximal degree of the Khovanov homology under twisting

本文研究了非零Khovanov同调群的最大同调度在扭曲下的性质。

Khovanov homology and binary dihedral representations for marked links

We introduce a version of Khovanov homology for alternating links with marking data, $ω$, inspired by instanton theory. We show that the analogue of the spectral sequence from Khovanov homology to singular instanton homology introduced in \cite{KM_unknot} for this marked Khovanov homology collapses on the $E_2$ page for alternating links. We moreover show that for non-split links the Khovanov homology we introduce for alternating links does not depend on $ω$; thus, the instanton homology also does not depend on $ω$ for non-split alternating links. Finally, we study a version of binary dihedral…

On links with Khovanov homology of small ranks

We classify all links whose Khovanov homology have ranks no greater than 8, and all three-component links whose Khovanov homology have ranks no greater than 12, where the coefficient ring is Z/2. The classification is based on the previous results of Kronheimer-Mrowka, Batson-Seed, Baldwin-Sivek, and the authors.

关于Khovanov同调与二重分支覆盖的注记

A remark on Khovanov homology and two-fold branched covers

结和链接的总秩的Khovanov同源性，但共享相同的两重分支覆盖区分的例子。因此，Khovanov同调不产生二重分支覆盖的不变量。

Khovanov homology and ribbon concordance

We show that a ribbon concordance between two links induces an injective map on Khovanov homology.

Khovanov homology detects the Hopf links

We prove that any link in $S^3$ whose Khovanov homology is the same as that of a Hopf link must be isotopic to that Hopf link. This holds for both reduced and unreduced Khovanov homology, and with coefficients in either $\mathbb{Z}$ or $\mathbb{Z}/2\mathbb{Z}$.

Homotopical Khovanov homology

We modify the definition of the Khovanov complex for oriented links in a thickening of an oriented surface to obtain a triply graded homological link invariant with a new homotopical grading.

平面Floer同调与2结的奇Khovanov同调

Plane Floer homology and the odd Khovanov homology of 2-knots

论文通过平面Floer同调证明了关于奇Khovanovcobordism映射的猜想，探索了奇Khovanov同调的 functorial 模型。

Khovanov同调的快速计算

Fast Khovanov Homology Computations

我们介绍了一个局部算法的Khovanov同调计算-也就是说，我们解释了它是如何可能的"取消"项在Khovanov复杂的相关（"本地"）缠结，从而取消了许多相关的"全球"条款在一个swoosh早期。这导致了显着提高计算效率。因此，我们的程序可以快速计算某些Khovanov同调群，否则将需要几个世纪来评估。

A generalized skein relation for Khovanov homology and a categorification of the $θ$-invariant

The Jones polynomial is a famous link invariant that can be defined diagrammatically via a skein relation. Khovanov homology is a richer link invariant that categorifies the Jones polynomial. Using spectral sequences, we obtain a skein-type relation satisfied by the Khovanov homology. Thanks to this relation, we are able to generalize the Khovanov homology in order to obtain a categorification of the $θ$-invariant, which is itself a generalization of the Jones polynomial.

Khovanov homology and exotic surfaces in the 4-ball

We show that the cobordism maps on Khovanov homology can distinguish smooth surfaces in the 4-ball that are exotically knotted (i.e., isotopic through ambient homeomorphisms but not ambient diffeomorphisms). We develop new techniques for distinguishing cobordism maps on Khovanov homology, drawing on knot symmetries and braid factorizations. We also show that Plamenevskaya's transverse invariant in Khovanov homology is preserved by maps induced by positive ascending cobordisms.

Hyperbolic Knots and Torsion in Khovanov Homology

In this note, we show that if there is a knot in $S^3$ having $\mathbb{Z}_m$ torsion in its Khovanov homology, then there are infinitely many hyperbolic knots and infinitely many prime satellite knots having $\mathbb{Z}_m$ torsion in their Khovanov homology. As an application, we give the first known examples of hyperbolic knots and links with odd (and other non-$\mathbb{Z}_2$ torsion) in their Khovanov homology.

On odd torsion in even Khovanov homology

This short note resolves the most important part of the PS braid conjecture while introducing the first known examples of knots and links with odd torsion of order 9, 27, 81, and 25 in their even Khovanov homology.

Normal surfaces and colored Khovanov homology

We show that colored Khovanov homology detects classes of essential surfaces as a direct analogue of the slope conjectures for the colored Jones polynomial. We do this by identifying certain generators of the colored Khovanov chain complex with normal surfaces in the complement of the knot using an ideal triangulation from a diagram.

A scanning algorithm for odd Khovanov homology

We adapt Bar-Natan's scanning algorithm for fast computations in (even) Khovanov homology to odd Khovanov homology. We use a mapping cone construction instead of a tensor product, which allows us to deal efficiently with the more complicated sign assignments in the odd theory. The algorithm has been implemented in a computer program. We also use the algorithm to determine the odd Khovanov homology of 3-strand torus links.

Patterns in Khovanov link and chromatic graph homology

Khovanov homology of a link and chromatic graph homology are known to be isomorphic in a range of homological gradings that depend on the girth of a graph. We discuss patterns shared by these two homology theories. In particular, we improve the bounds for the homological span of chromatic homology by Helme-Guizon, Przytycki and Rong. An explicit formula for the rank of the third chromatic homology group on the main diagonal is given and used to compute the corresponding Khovanov homology group and the fourth coefficient of the Jones polynomial for links with certain diagrams.

环形Khovanov同调与增广链环

Annular Khovanov homology and augmented links

本文构造了从环形Khovanov同调收敛至增广链环约化Khovanov同调的谱序列，并据此分类了极小秩链环，证明了该同调能检测分离链环。

On the maximal degree of the Khovanov homology

It is known that the maximal homological degree of the Khovanov homology of a knot gives a lower bound of the minimal positive crossing number of the knot. In this paper, we show that the maximal homological degree of the Khovanov homology of a cabling of a knot gives a lower bound of the minimal positive crossing number of the knot.

Khovanov同调与奇异瞬子同调中的柄映射 I

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。

霍万诺夫同调与卷旋切片圆盘

Khovanov homology and roll-spun slice disks

研究霍万诺夫同调对卷旋切片圆盘的区分能力，通过与莫尔斯1-柄合成进行研究，发现其无法区分特定的两种切片圆盘。

Ribbon distance and Khovanov homology

We study a notion of distance between knots, defined in terms of the number of saddles in ribbon concordances connecting the knots. We construct a lower bound on this distance using the X-action on Lee's perturbation of Khovanov homology.

某些有限填充结果的Khovanov同调新证明

New proofs of certain finite filling results via Khovanov homology

给出了双曲扭纽结不允许非平凡Dehn变换的Khovanov同调性证明，证明了有限基本群的双曲扭纽结不允许非平凡Dehn变换。

Khovanov同调的纽结

Knots with identical Khovanov homology

我们给出了一个配方，用于构建家庭的不同的结，具有相同的Khovanov同源性，并给出了对总理结的例子，以及无限的家庭，这个属性。

On Khovanov Homology and Related Invariants

This paper begins with a survey of some applications of Khovanov homology to low-dimensional topology, with an eye toward extending these results to $\mathfrak{sl}(n)$ homologies. We extend Levine and Zemke's ribbon concordance obstruction from Khovanov homology to $\mathfrak{sl}(n)$ homology for $n \geq 2$, including the universal $\mathfrak{sl}(2)$ and $\mathfrak{sl}(3)$ homology theories. Inspired by Alishahi and Dowlin's bounds for the unknotting number coming from Khovanov homology and relying on spectral sequence arguments, we produce bounds on the alternation number of a knot. Lee and B…

Concordance invariants from the $E(-1)$ spectral sequence on Khovanov homology

We construct the concordance invariant coming from the $E(-1)$ spectral sequence on Khovanov homology in the same way Rasmussen's $s$ invariant comes from the Lee spectral sequence, and show that it gives a bound on the nonorientable slice genus.

Rotors in Khovanov Homology

Anstee, Przyticki and Rolfsen introduced the idea of rotants, pairs of links related by a generalised form of link mutation. We exhibit infinitely many pairs of rotants which can be distinguished by Khovanov homology, but not by the Jones polynomial.

Khovanov同调论及其应用

Khovanov homology theories and their applications

这是一个临时文件讨论各种版本的霍瓦诺夫同调论，它们之间的相互关系，它们的性质，以及它们的应用到其他领域的纽结理论和低维拓扑。

Khovanov同源性是一个未结探测器

Khovanov homology is an unknot-detector

我们证明了一个纽结是unknot当且仅当它的约化Khovanov上同调秩为1。证明有两个步骤。首先，我们表明，有一个谱序列开始减少Khovanov上同调和邻接的结使用奇异瞬子定义的同调。然后，我们表明，后者的同源性是同构的瞬子弗洛尔同调的缝合结补：一个不变的，是已知的检测unknot。

索环的Khovanov同调的最大次数

The maximal degree of the Khovanov homology of a cable link

本文研究了索环的Khovanov同调。我们首先估计了（$2k+1$，$(2k+1)n$）-环面环的Khovanov同调的最大同调度项，并给出了其同调厚度的下界.具体地说，我们表明，同源厚度的（$2k+1$，$(2k+1)n$）环面链接大于或等于$k^{2}n+2$。接下来，我们研究了具有足够大的$n$的任何结的（$p$，$pn$）-成缆的Khovanov同源性的最大同源度。此外，我们还计算了这样一个链接甚至$p$的Khovanov同调的最大同调度项。作为一个应用，我们计算的Khovanov同调和Rasmussen不变量的扭曲的Whitehead双重的任何结有足够多的扭曲。

Khovanov homology and knot Floer homology for pointed links

A well-known conjecture states that for any $l$-component link $L$ in $S^3$, the rank of the knot Floer homology of $L$ (over any field) is less than or equal to $2^{l-1}$ times the rank of the reduced Khovanov homology of $L$. In this paper, we describe a framework that might be used to prove this conjecture. We construct a modified version of Khovanov homology for links with multiple basepoints and show that it mimics the behavior of knot Floer homology. We also introduce a new spectral sequence converging to knot Floer homology whose $E_1$ page is conjecturally isomorphic to our new version…

3-strand pretzel链的有理Khovanov同调

The rational Khovanov homology of 3-strand pretzel links

3股椒盐卷饼结和链接是纽结理论中一个经过充分研究的例子来源。然而，虽然已经计算了3股椒盐卷饼结的一些子家族的Khovanov同源性，但没有给出所有这些的通用公式。我们给出了一个一般公式的未约Khovanov同源的所有3链椒盐卷饼链接，在有理数。

Search for torsion in Khovanov homology

In the Khovanov homology of links, presence of $\mathbb{Z}_2$-torsion is a very common phenomenon. Finite number of examples of knots with $\mathbb{Z}_n$-torsion for $n>2$ were also known, none for $n>8$. In this paper, we prove that there are infinite families of links whose Khovanov homology contains $\mathbb{Z}_n$-torsion for $2 < n < 9$ and $\mathbb{Z}_{2^s}$-torsion for $s < 24$. We also introduce $4$-braid links with $\mathbb{Z}_3$-torsion which are counterexamples to the PS braid conjecture. We also provide an infinite family of knots with $\mathbb{Z}_5$-torsion in reduced Khovanov homo…

Khovanov homology and the symmetry group of a knot

We introduce an invariant of tangles in Khovanov homology by considering a natural inverse system of Khovanov homology groups. As application, we derive an invariant of strongly invertible knots; this invariant takes the form of a graded vector space that vanishes if and only if the strongly invertible knot is trivial. While closely tied to Khovanov homology -- and hence the Jones polynomial -- we observe that this new invariant detects non-amphicheirality in subtle cases where Khovanov homology fails to do so. In fact, we exhibit examples of knots that are not distinguished by Khovanov homolo…

Khovanov同调与交错纽结的扭数

Khovanov Homology and the Twist Number of Alternating Knots

X.S. Lin和O.达斯巴赫证明了交替纽结的琼斯多项式的第二个和倒数第二个系数的绝对值之和等于纽结的捻数。本文利用Khovanov同调给出了其结果的一个新的证明。证明是通过归纳的交叉数使用长的准确序列在Khovanov同调对应的考夫曼括号绞关系。

与对称环相关的等变Khovanov同调

Equivariant Khovanov homology associated with symmetric links

设$Δ$是三球面中的平凡纽结。对每一个奇数阶有限循环群$G$，构造了系数在域$\F_{2}$中的$G$ -等变Khovanov同调.这种同调是直到$(S^{3},Δ)$中的合痕的连接的不变量。另一种解释是使用分类的考夫曼括号绞模的固体环面。我们的技术也适用于图的情况，以定义图同调的等变版本，该版本将色多项式归类。关键词：霍瓦诺夫同调，群作用，等变琼斯多项式，绞模。