John A. Baldwin, Nathan Dowlin, Adam Simon Levine, Tye Lidman, Radmila Sazdanovic
Using Dowlin's spectral sequence from Khovanov homology to knot Floer homology, we prove that reduced Khovanov homology (over $\mathbb{Q}$) detects the figure-eight knot.
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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。
Genus two mutant knots with the same dimension in knot Floer and Khovanov homologies
我们展示了一个无限族的同构纽结Heegaard弗洛尔同调。在这个无限的家庭中的每一个结承认一个非平凡的亏格两个突变体，共享相同的总尺寸在两个结弗洛尔同调和Khovanov同源。每一个纽结通过纽结的弗洛尔同调和Khovanov同调作为双阶群与其属的两个突变体区分开来。此外，对于纽结Heegaard弗洛尔同调性和Khovanov同源性，两种突变互换了$δ$分级$k$和$-k$中的组.
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.
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。
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.
In this paper we compute the signature for a family of knots $W(k,n)$, the weaving knots of type $(k,n)$. By work of E.~S.~Lee the signature calculation implies a vanishing theorem for the Khovanov homology of weaving knots. Specializing to knots $W(3,n)$, we develop recursion relations that enable us to compute the Jones polynomial of $W(3,n)$. Using additional results of Lee, we compute the ranks of the Khovanov Homology of these knots. At the end we provide evidence for our conjecture that, asymptotically, the ranks of Khovanov Homology of $W(3,n)$ are {\it normally distributed}.
An infinite family of strongly invertible L-space knots without Khovanov thin surgery
本文针对已知仅存3个强可逆L空间纽结无Khovanov薄手术的情况，构造出此类纽结的首个无限族，丰富了相关纽结理论的研究成果。
Khovanov-Rozansky homology and the braid index of a knot
我们证明了一个结的存在，其辫子指数Morton-Franks-Williams不等式未能检测到，但相关的不等式（KR-MFW不等式），它使用了新的信息Khovanov-Rozansky同源，检测。我们还证明，通过例子，存在无穷多个结的KR-MFW不等式未能检测到的辫子指数。
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。
The Kanenobu knots and Khovanov-Rozansky homology
Kanenobu已经给出了具有相同HOMFLY多项式的无穷多个节点。我们表明，这些结也有相同的sl（n）和HOMFLY同源，从而给出了第一个例子的一个无限的家庭结无法区分这些不变量。这是一个关于结的同调的结构定理的结果，通过用一条丝带缠绕一个丝带结的丝带而得到。
For a connected cobordism S between two knots K1,K2 in S3, we establish an inequality involving the number of local maxima, the genus of S, and the torsion orders of Kht(K1),Kht(K2), where Kht denotes Lee's perturbation of Khovanov homology. This shows that the torsion order gives a lower bound for the band-unlinking number.
Khovanov Homology, Lee Homology and a Rasmussen Invariant for Virtual Knots
本文包含了一个基本上自足的治疗Khovanov同调，Khovanov-Lee同调以及Rasmussen不变量的虚拟结和虚拟结配边，直接适用于经典的结和经典的结配边。要做到这一点，我们给出了一个替代配方的Manturov定义的Khovanov同调的虚拟结和链接与任意系数。这种方法使用纽结图上的切割轨迹来诱导Frobenius代数中的共轭算子。然后，我们讨论的影响，在上述理论的通用Frobenius代数的虚拟节点诱导的地图。接下来，我们将展示如何应用Karoubi包络的方法Bar-Natan和莫里森抽象的链接图与横截构建规范的Khovanov-李同源发电机。使用这些典型的发电机，我们推导出一个推广的Rasmussen不变量的虚拟结配边，并进一步推广Rasmussen的结果切片属正结的情况下，积极的虚拟结。还应该注意的是，这种推广的拉斯穆森不变量提供了一个很容易计算的障碍，在$S_g \times I \times I$的纽结配边在Turaev的意义。
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.
Involutive Khovanov homology and equivariant knots II
本文将对合Khovanov同调框架推广到对合缠结，实现了相关同调与不变量的分治加速计算，并得到存在奇异切片圆盘对的强可逆纽结新例子。
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。
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…
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…
State Cycles, Quasipositive Modification, and Constructing H-thick Knots in Khovanov Homology
我们研究了Khovanov同调类的状态循环代表，并研究他们如何与Jacobsson同态和李的地图$Φ$。作为一个应用，我们描述了一个一般的程序，拟正修改，用于构建H厚结的合理Khovanov同调。此外，我们表明，这种结的特定家庭不能检测到Khovanov的厚度标准。我们还展示了一系列的准正修改相关的主要环节的宽度增加。
A module structure on odd Khovanov homology and the odd invariant for ribbon 2-knots
证明链环\(L\)的约化奇数霍万诺夫同调自然是其分支双覆盖的第一同调的外代数上的模，从几何角度描述该模结构并与链环协边诱导的奇数霍万诺夫映射相关，还给出相关组合证明。
Khovanov concordance minima and the (4,5) torus knot
研究了 (4,5) 螺旋结的 Khovanov homology 在 concordance 类中的性质，证明其为该类中任意结的 Khovanov homology 的 summand。
Ozsvath and Szabo recently constructed an algebraically defined invariant of tangles which takes the form of a DA bimodule. This invariant is expected to compute knot Floer homology. The authors have a similar construction for open braids and their plat closures which can be viewed as a filtered DA bimodule over the same algebras. For a closed diagram, this invariant computes the Khovanov homology of the knot or link. We show that forgetting the filtration, our DA bimodules are homotopy equivalent to a suitable version of the Ozsvath- Szabo bimodules. In addition to giving a relationship betwe…
Involutive Khovanov homology and equivariant knots
本文提出了一种involutive Khovanov同调，用于研究强不可约结的exotic slice disks性质，并重新证明了Hayden的无限族结J_n的exotic性。
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。
Spectra in Khovanov and knot Floer theories
本文介绍 Khovanov 理论与纽结 Floer 理论中的稳定同伦类型，内容源自 Sarkar 在 Rényi 研究所的系列讲座。
A Khovanov Laplacian and Khovanov Dirac for Knots and Links
本研究引入霍万诺夫拉普拉斯算子与霍万诺夫狄拉克算子研究纽结和链环图，其调和谱保留Khovanov同调的拓扑不变量，非调和谱可提供Khovanov同调之外的新拓扑信息。
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。
Simplified Khovanov-Rozansky Chain Complexes of Open 2-braids
受Krasner [arXiv：0801.4018]和Lobb [arXiv：1103.1412]工作的启发，我们简化了开2-辫子的Khovanov-Rozansky链复形.作为一个应用，我们表明，对于一个结包含一个“长”2-辫子，SL（N）拉斯穆森不变量这个结线性依赖于这个2-辫子的长度。我们改进了（2，2k+1）索结的结果，作为一个简单的推论，计算了（2，2k+1）条索和双螺旋索结的sl（2）Rasmussen不变量.
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。
The support of the Khovanov's invariants for alternating knots
在这篇文章中，我们证明了Bar-Natan猜想，Garoufaldine，和Khovanov的支持Khovanov的不变量交替结。
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.
In this paper we introduce a chain complex $C_{1 \pm 1}(D)$ where D is a plat braid diagram for a knot K. This complex is inspired by knot Floer homology, but it the construction is purely algebraic. It is constructed as an oriented cube of resolutions with differential d=d_0+d_1. We show that the E_2 page of the associated spectral sequence is isomorphic to the Khovanov homology of K, and that the total homology is a link invariant which we conjecture is isomorphic to δ-graded knot Floer homology. The complex can be refined to a tangle invariant for braids on 2n strands, where the associated …
Conjectures on the Khovanov Homology of Torus Knots, Twist Knots, and Legendrian Simple Knots
本文基于Khovanov同调的性质，提出涡旋结和扭结的同调能区分所有Legendrian简单结的猜想，并通过大量数据验证其有效性。
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。
Khovanov homology of positive links and of L-space knots
本文利用Khovanov同调建立正链环的新障碍，证明第一同调群消失等价于纤维化，并推广至缆绳及L-空间纽结。
Khovanov-Jacobsson numbers and invariants of surface-knots derived from Bar-Natan's theory
Khovanov介绍了一个上同调理论的定向经典链接的分级欧拉示性数是琼斯多项式。由于Khovanov的理论是函子的经典链接之间的链接配边，我们得到了一个不变量的表面结，称为{\it Khovanov-Jacobsson数}，通过考虑表面结之间的链接配边空链接。本文利用Bar-Natan理论定义了Khovanov-Jacobsson数的推广--曲面纽结的不变量，并证明了任何$T^2$-纽结都有平凡的Khovanov-Jacobsson数.
The (untwisted) oriented cube of resolutions for knot Floer homology assigns a complex $C_{F}(S)$ to a singular resolution $S$ of a knot $K$. Manolescu conjectured that when $S$ is in braid position, the homology $H_{*}(C_{F}(S))$ is isomorphic to the HOMFLY-PT homology of $S$. Together with a naturality condition on the induced edge maps, this conjecture would prove the spectral sequence from HOMFLY-PT homology to knot Floer homology. Using a basepoint filtration on $C_{F}(S)$, a recursion formula for HOMFLY-PT homology, and additional $sl_{n}$-like differentials on $C_{F}(S)$, we prove this …
A slice genus lower bound from sl(n) Khovanov-Rozansky homology
我们证明，扰动sl（n）Khovanov-Rozansky链接同调的定义给出了结的切片亏格的下界。作为一个推论，这产生另一个证明米尔诺猜想片属的环面结。
Khovanov-Rozansky homology over $\mathbb{F}[U,V]$
本文定义了F[U,V]上的完全Khovanov-Rozansky同调链复形CH(K)，证明其链同伦型为纽结不变量，并利用该结构定义了纽结Floer配边不变量τ、ε、φ_j在Khovanov-Rozansky框架下的对应物。
We construct an algebra of non-trivial homological operations on Khovanov homology with coefficients in $\mathbb Z_2$ generated by two Bockstein operations. We use the unified Khovanov homology theory developed by the first author to lift this algebra to integral Khovanov homology. We conjecture that these two algebras are infinite and present evidence in support of our conjectures. Finally, we list examples of knots that have the same even and odd Khovanov homology, but different actions of these homological operations. This confirms that the unified theory is a finer knot invariant than the …
Ribbon-moves for 2-knots with 1-handles attached and Khovanov-Jacobsson numbers
证明了2-纽结上沿二重点圆的交叉变换是由2-纽结附加1-柄得到的打结环面的带状移动实现的。因此，任何2-纽结，其交叉变化是一个unknotting操作，如带状2-纽结和twist-spun结，有平凡的Khovanov-Jacobsson数。
Khovanov-Rozansky homology of two-bridge knots and links
我们计算的减少版本的Khovanov和Rozansky的sl（N）的同源性的两个桥梁结和链接。答案用HOMFLY多项式和签名表示。
Khovanov homology of strongly invertible knots and their quotients
本文通过构造联系强可逆纽结与其商纽结的谱序列，重新证明Khovanov同调可区分某些切片圆盘，并给出分支二重覆盖的Heegaard Floer同调类比结果。
A new computational tool for Khovanov cobordism maps
本文提出了一种新的Python工具，用于计算Khovanov同调上的cobordism映射，并应用于区分由边界结对称性产生的ribbon盘。
By considering a version of Khovanov homology incorporating both the Lee and $E(-1)$ differentials, we construct a $1$-parameter family of concordance homomorphisms similar to the Upsilon invariant from knot Floer homology. This invariant gives lower bounds on the slice genus and can be used to prove that certain infinite families of pretzel knots are linearly independent in the smooth concordance group.
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.
On spectral sequences from Khovanov homology
有一些同调结不变量，每一个满足一个无方向的绞合精确序列，它可以实现为一个谱序列的极限页开始于一个版本的Khovanov链复形。基本的1-手柄电影移动的合成诱导谱序列的态射。这些态射在文献中仍然没有被利用，也许是因为关于一般电影所诱导的地图的自然性仍然存在一个悬而未决的问题。本文主要研究了由Kronheimer-Mrowka提出的从Khovanov同调到瞬子结弗洛尔同调的谱序列，以及由Ozsváth-Szabó提出的从Heegaard-弗洛尔同调到分支复盖的谱序列.例如，我们使用1-柄态射给出了关于瞬子结和（4，5）环面结的弗洛尔同调的滤子的新信息，确定了这些滤子直到一对度中的模糊性;确定了无限类素结的Ozsváth-Szabó谱序列;并表明Kronheimer-Mrowka和Ozsváth-Szabó谱序列的较高微分必然会降低所有椒盐卷饼结的δ分级。
A minimality property for knots without Khovanov 2-torsion
针对 Shumakovitch 关于非平凡纽结 Khovanov 同调 2-挠的猜想，本文证明无 2-挠纽结在真有理缠结替换下约化 Khovanov 同调秩的极小性，并推得解结数 1 的纽结必有 2-挠。
We define a homology theory of virtual links built out of the direct sum of the standard Khovanov complex with itself, motivating the name doubled Khovanov homology. We demonstrate that it can be used to show that some virtual links are non-classical, and that it yields a condition on a virtual knot being the connect sum of two unknots. Further, we show that doubled Khovanov homology possesses a perturbation analogous to that defined by Lee in the classical case and define a doubled Rasmussen invariant. This invariant is used to obtain various cobordism obstructions; in particular it is an obs…
Legendrian simple knots not detected by Khovanov homology
本研究构造了Khovanov同调无法检测的Legendrian简单纽结，推翻了Chernov和Maguire的相关猜想。
Does Khovanov homology detect the unknot?
我们确定了一个广泛的类的结，其中包括解开号码1结，其中Khovanov同源检测解开。一个推论是，许多卫星结的Khovanov同源性，包括怀特黑德双，检测unknot。
On Khovanov's cobordism theory for su(3) knot homology
本文重新考虑了Khovanov在Math.QA/0304375中定义的su（3）链同调论，以及Mackaay和Vaz在Math.GT/0603307中推广的su（3）链同调论.稍加修改后，我们将该理论描述为从缠结的平面代数到“有缝协边”（实际上是“cobordisms with seams”）（复合体）的平面代数（实际上是一个“canopolis”）的映射，使其在Bar-Natan的局部su（2）数学理论的意义上成为局部的。GT/0410495。我们表明，这种“有缝协边主义canopolis”decegorifies给正是你希望和期望：Kuperberg的su（3）蜘蛛在q-alg/9712003中定义。我们猜想一个答案，一个更有趣的问题，我们的配边理论的Karoubi信封的decategorification。最后，我们描述了如何理论实际上是完全可计算的，并给出了（2，n）环面结的su（3）同调的详细计算。
Transverse knots and Khovanov homology
我们定义了一个不变量的横向链接在标准的接触3-球面作为一个杰出的元素的Khovanov同源的链接。这个不变量的量子分次就是链环的自环绕数。对于纽结，这给出了关于自环绕数的Rasmussen不变量s（K）的界。我们证明了我们的不变量消失的横向结稳定，它是非零的准正辫子。我们还讨论了Heegaard Floer不变量的连接。
This paper is an extended account of my "Introductory Plenary talk at Knots in Hellas 2016" conference We start from the short introduction to Knot Theory from the historical perspective, starting from Heraclas text (the first century AD), mentioning R.Llull (1232-1315), A.Kircher (1602-1680), Leibniz idea of Geometria Situs (1679), and J.B.Listing (student of Gauss) work of 1847. We spend some space on Ralph H. Fox (1913-1973) elementary introduction to diagram colorings (1956). In the second section we describe how Fox work was generalized to distributive colorings (racks and quandles) and e…
O. Plamenevskaya associated to each transverse knot K an element of the Khovanov homology of K. In this paper, we give two refinements of Plamenevskaya's invariant, one valued in Bar-Natan's deformation of the Khovanov complex and another as a cohomotopy element of the Khovanov spectrum. We show that the first of these refinements is invariant under negative flypes and SZ moves; this implies that Plamenevskaya's class is also invariant under these moves. We go on to show that for small-crossing transverse knots K, both refinements are determined by the classical invariants of K.
We prove that every $\mathbb{Z}_2$H-thin link has no $2^k$-torsion for $k>1$ in its Khovanov homology. Together with previous results by Eun Soo Lee and the author, this implies that integer Khovanov homology of non-split alternating links is completely determined by the Jones polynomial and signature. Our proof is based on establishing an algebraic relation between Bockstein and Turner differentials on Khovanov homology over $\mathbb{Z}_2$. We conjecture that a similar relation exists between the corresponding spectral sequences.
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…
On stable Khovanov homology of torus knots
我们猜想环面纽结的稳定Khovanov同调可以描述为显式非正则序列的二次多项式的Koszul同调。相应的庞加莱级数与罗杰斯-拉马努金恒等式有关。
Khovanov homology of tangles: algorithm and computation
针对缠结的霍瓦诺夫同调计算未充分探索的问题，提出拓扑量子场论构造及综合计算方法，含详细步骤与代码实现指南，推动该领域研究与应用。
We utilize relations between Khovanov and chromatic graph homology to determine extreme Khovanov groups and corresponding coefficients of the Jones polynomial. The extent to which chromatic homology and chromatic polynomial can be used to compute integral Khovanov homology of a link depends on the maximal girth of its all-positive graphs. In this paper we also define the girth of a link, discuss relations to other knot invariants, and the possible values for girth. Analyzing girth leads to a description of possible all-A state graphs of any given link; e.g., if a link has a diagram such that t…
Persistent Khovanov homology of tangles
本文提出缠结的持续Khovanov同调框架，通过具体函子和平面代数构造，解决现有方法难以捕捉纽结与链环局部拓扑特征的问题，为曲线型数据拓扑分析提供理论基础与实践策略。
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