A refinement of Khovanov-Rozansky link homology

Some differentials on Khovanov-Rozansky homology

We study the relationship between the HOMFLY and sl(N) knot homologies introduced by Khovanov and Rozansky. For each N>0, we show there is a spectral sequence which starts at the HOMFLY homology and converges to the sl(N) homology. As an application, we determine the KR-homology of knots with 9 crossings or fewer.

A note on Gornik's perturbation of Khovanov-Rozansky homology

We show that the information contained in the associated graded vector space to Gornik's version of Khovanov-Rozansky knot homology is equivalent to a single even integer s_n(K). Furthermore we show that s_n is a homomorphism from the smooth knot concordance group to the integers. This is in analogy with Rasmussen's invariant coming from a perturbation of Khovanov homology.

关于有限特征下Khovanov–Rozansky同调的若干无穷小对称性的注记

Remarks on some infinitesimal symmetries of Khovanov--Rozansky homologies in finite characteristic

该研究给出有限特征下Khovanov–Rozansky同调基点无关性定理的新证明，还探讨了对应特征下gl(p)-同调的更多对称性。

锚定 $\mathfrak{gl}_N$ 泡沫与环带 Khovanov-Rozansky 同调

Anchored $\mathfrak{gl}_N$ foams and annular Khovanov-Rozansky homology

本文通过泡沫评估为加厚环带中的链环引入等变 $\mathfrak{gl}_N$ 同调，扩展了 Khovanov-Rozansky 理论。

Khovanov-Rozansky homology for infinite multi-colored braids

We define a limiting $\mathfrak{sl}_N$ Khovanov-Rozansky homology for semi-infinite positive multi-colored braids, and we show that this limiting homology categorifies a highest-weight projector for a large class of such braids. This effectively completes the extension of Cautis' similar result for infinite twist braids, begun in our earlier papers with Islambouli and Abel. We also present several similar results for other families of semi-infinite and bi-infinite multi-colored braids.

带色Khovanov-Rozansky同调的对称性

Symmetries of colored Khovanov-Rozansky homology

该研究在带色$\frak{gl}_N$-Khovanov-Rozansky同调上构造Witt代数正半部分的作用，证明其与带色链环同调的函子性相容，借助Bar-Natan的范畴框架在缠结层面完成相关命题的证明。

Colored Khovanov-Rozansky homology for infinite braids

We show that the limiting unicolored $\mathfrak{sl}(N)$ Khovanov-Rozansky chain complex of any infinite positive braid categorifies a highest-weight projector. This result extends an earlier result of Cautis categorifying highest-weight projectors using the limiting complex of infinite torus braids. Additionally, we show that the results hold in the case of colored HOMFLY-PT Khovanov-Rozansky homology as well. An application of this result is given in finding a partial isomorphism between the HOMFLY-PT homology of any braid positive link and the stable HOMFLY-PT homology of the infinite torus …

Symmetric Khovanov--Rozansky link homologies

We provide a finite dimensional categorification of the symmetric evaluation of $\mathfrak{sl}_N$-webs using foam technology. As an output we obtain a symmetric link homology theory categorifying the link invariant associated to symmetric powers of the standard representation of $\mathfrak{sl}_N$. In addition, the construction is actually made in an equivariant setting. We prove also that there is a spectral sequence from the Khovanov-Rozansky triply graded link homology to the symmetric one and provide along the way a foam interpretation of Soergel bimodules.

Khovanov-Rozansky同调中的平凡类与对称性

Tautological classes and symmetry in Khovanov-Rozansky homology

该研究在Khovanov-Rozansky链环同调中定义新交换算子族$F_k$，证明$F_2$满足硬莱夫谢茨性质，验证了相关学者提出的同调对称性猜想。

Witt 代数在 Khovanov-Rozansky 同调上的作用

An action of the Witt algebra on Khovanov-Rozansky homology

本文构造了 Witt 代数正部分在 Khovanov-Rozansky gl_N 链环同调上的作用，并证明链环配边诱导同调的扭转之间的等变映射，同时将简单网的状态空间等同于多项式上的标准 Witt 代数表示，由此推导出与 Lee 同调和亏格界的简单关系。

Computing Khovanov-Rozansky homology and defect fusion

We compute the categorified sl(N) link invariants as defined by Khovanov and Rozansky, for various links and values of N. This is made tractable by an algorithm for reducing tensor products of matrix factorisations to finite rank, which we implement in the computer algebra package Singular.

Universal Khovanov-Rozansky sl(2) cohomology

We generalize the Khovanov-Rozansky cohomology for n=2 by means of a homogeneous potential that depends on two parameters, to obtain the universal Khovanov-Rozansky sl(2) link cohomology. This theory is equivalent to the universal foam sl(2) link cohomology, after tensoring both theories with appropriate rings.

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…

Invariants of 4-manifolds from Khovanov-Rozansky link homology

We use Khovanov-Rozansky gl(N) link homology to define invariants of oriented smooth 4-manifolds, as skein modules constructed from certain 4-categories with well-behaved duals. The technical heart of this construction is a proof of the sweep-around property, which makes these link homologies well defined in the 3-sphere.

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.

A slice genus lower bound from sl(n) Khovanov-Rozansky homology

We show that perturbing the definition of sl(n) Khovanov-Rozansky link homology gives a lower bound on the slice genus of a knot. As a corollary this yields another proof of Milnor's conjecture on the slice genus of torus knots.

Khovanov-Rozansky homology and the braid index of a knot

We prove the existence of a knot whose braid index the Morton-Franks-Williams inequality fails to detect but a related inequality (KR-MFW inequality), which uses new information of Khovanov-Rozansky homology, detects. We also prove, by examples, that there exists infinitely many knots for which the KR-MFW inequality fails to detect the braid indices.

F[U,V]上的Khovanov-Rozansky同调

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。

The Khovanov-Rozansky Cohomology and Bennequin Inequalities

We review Bennequin type inequalities established using various versions of the Khovanov-Rozansky cohomology. Then we give a new proof of a Bennequin type inequality established by the author, and derive new Bennequin type inequalities for knots using Gornik's version of the Khovanov-Rozansky cohomology, which generalize those established by Shumakovitch, Plamenevskaya and Kawamura using the Rasmussen invariant.

约化的Khovanov-Rozansky同调

Reduced Khovanov-Rozansky homology

本文定义了对应股线间区域变量的约化Khovanov-Rozansky同调变体，重建了Khovanov与Rasmussen方法等价等民间结果，还证明Rasmussen链复形是Rouquier链复形的自由分解。

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

In this note we present an explicit isomorphism between Khovanov-Rozansky $sl_2$-homology and ordinary Khovanov homology. This result was originally stated in Khovanov and Rozansky's paper \cite{KRI}, though the details have yet to appear in the literature. The main missing detail is providing a coherent choice of signs when identifying variables in the $sl_2$-homology. Along with the behavior of the signs and local orientations in the $sl_2$-homology, both theories behave differently when we try to extend their definitions to virtual links, which seemed to suggest that the $sl_2$-homology may…

2-Verma modules and the Khovanov-Rozansky link homologies

We explain how Queffelec-Sartori's construction of the HOMFLY-PT link polynomial can be interpreted in terms of parabolic Verma modules for $\mathfrak{gl}_{2n}$. Lifting the construction to the world of categorification, we use parabolic 2-Verma modules to give a higher representation theory construction of Khovanov-Rozansky's triply graded link homology.

Khovanov-Rozansky homology via a canopolis formalism

In this paper, we describe a canopolis (i.e. categorified planar algebra) formalism for Khovanov and Rozansky's link homology theory. We show how this allows us to organize simplifications in the matrix factorizations appearing in their theory. In particular, it will put the equivalence of the original definition of Khovanov-Rozansky homology and the definition using Soergel bimodules in a more general context, allow us to give a new proof of the invariance of triply graded homology and give new analysis of the behavior of triply graded homology under the Reidemeister IIb move.

The $\mathfrak{sl}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality

We give a purely combinatorial construction of colored $\mathfrak{sl}_n$ link homology. The invariant takes values in a 2-category where 2-morphisms are given by foams, singular cobordisms between $\mathfrak{sl}_n$ webs; applying a (TQFT-like) representable functor recovers (colored) Khovanov-Rozansky homology. Novel features of the theory include the introduction of `enhanced' foam facets which fix sign issues associated with the original matrix factorization formulation and the use of skew Howe duality to show that (enhanced) closed foams can be evaluated in a completely combinatorial manner…

Khovanov-Rozansky同调上的翻转对称性

The flip symmetry on Khovanov-Rozansky homology

本文证明了链环图翻转对称性在Khovanov-Rozansky ${\frak{gl}}_{N}$同调上诱导的对合可对角化且特征值为$\pm1$，并利用同伦扰动和Soergel双模图解方法完成了证明。

Serre duality for Khovanov-Rozansky homology

We prove that the full twist is a Serre functor in the homotopy category of type A Soergel bimodules. As a consequence, we relate the top and bottom Hochschild degrees in Khovanov-Rozansky homology, categorifying a theorem of Kálmán.

Simplified Khovanov-Rozansky Chain Complexes of Open 2-braids

Motivated by the works of Krasner [arXiv:0801.4018] and Lobb [arXiv:1103.1412], we simplify the Khovanov-Rozansky chain complexes of open 2-braids. As an application, we show that, for a knot containing a "long" 2-braid, the sl(N) Rasmussen invariant of this knot depends linearly on the length of this 2-braid. We refine this result for (2,2k+1) cable knots and, as a simple corollary, compute the sl(2) Rasmussen invariants of (2,2k+1) cables of slice and amphicheiral knots.

A cobordism category attached to Khovanov-Rozansky link homologies based on operads

We consider colored operads and their actions on categories. As a special example we construct a cobordism category with a colored operad action arising from oriented planar arc diagrams. This is used to construct an invariant of oriented tangle diagrams with values in the homotopy category attached to the cobordism category. Motivated by Bar-Natan's categorification of the Jones polynomial, it categorifies the quantum $sl_n$ quantum invariants and is adapted to the categorification of the $sl_n$ quantum invariants by Khovanov and Rozansky using matrix factorizations. We conjecture to exist th…

Equivariant Khovanov-Rozansky Homology and Lee-Gornik Spectral Sequence

Lobb observed in [arXiv:1103.1412] that each equivariant sl(N) Khovanov-Rozansky homology over C[a] admits a standard decomposition of a simple form. In the present paper, we derive a formula for the corresponding Lee-Gornik spectral sequence in terms of this decomposition. Based on this formula, we give a simple alternative definition of the Lee-Gornik spectral sequence using exact couples. We also demonstrate that an equivariant sl(N) Khovanov-Rozansky homology over C[a] can be recovered from the corresponding Lee-Gornik spectral sequence via this formula. Therefore, these two algebraic inva…

On the quantum filtration of the Khovanov-Rozansky cohomology

We prove the quantum filtration on the Khovanov-Rozansky link cohomology H_p with a general degree (n+1) monic potential polynomial p(x) is invariant under Reidemeister moves, and construct a spectral sequence converging to H_p that is invariant under Reidemeister moves, whose E_1 term is isomorphic to the Khovanov-Rozansky sl(n)-cohomology H_n. Then we define a generalization of the Rasmussen invariant, and study some of its properties. We also discuss relations between upper bounds of the self-linking number of transversal links in the standard contact three sphere.

Link splitting deformation of colored Khovanov--Rozansky homology

We introduce a multi-parameter deformation of the triply-graded Khovanov--Rozansky homology of links colored by one-column Young diagrams, generalizing the "$y$-ified" link homology of Gorsky--Hogancamp and work of Cautis--Lauda--Sussan. For each link component, the natural set of deformation parameters corresponds to interpolation coordinates on the Hilbert scheme of the plane. We extend our deformed link homology theory to braids by introducing a monoidal dg 2-category of curved complexes of type $A$ singular Soergel bimodules. Using this framework, we promote to the curved setting the categ…

The Kanenobu knots and Khovanov-Rozansky homology

Kanenobu has given infinite families of knots with the same HOMFLY polynomials. We show that these knots also have the same sl(n) and HOMFLY homologies, thus giving the first example of an infinite family of knots undistinguishable by these invariants. This is a consequence of a structure theorem about the homologies of knots obtained by twisting up the ribbon of a ribbon knot with one ribbon.

Khovanov-Rozansky 同调、Bott-Samelson 空间与扭曲上同调

Khovanov-Rozansky homologies, Bott-Samelson spaces and twisted cohomology

本文比较了 Rasmussen 的 Khovanov-Rozansky 同调表述与 Kitchloo 基于 Bott-Samelson 簇和 Borel 等变上同调扭曲的链环不变量，证明这些几何构造可产生具有特殊化势或万有势的等变整系数 sl(n) 链环同调。

Braids, Transversal links and the Khovanov-Rozansky Theory

We establish some inequalities about the Khovanov-Rozansky cohomologies of braids. These give new upper bounds of the self-linking numbers of transversal links in standard contact $S^3$ which is sharper than the well known bound given by the HOMFLY polynomial. We also introduce a sequence of transversal link invariants, and discuss some of their properties.

等变Khovanov-Rozansky同调的对称性

Symmetries of equivariant Khovanov-Rozansky homology

本文在等变$\mathfrak{gl}_N$-链环同调上构造$\mathfrak{sl}_2$-作用，进而得到$p$-DG结构，并探索其拓扑应用。

On the Transverse Khovanov-Rozansky Homologies: Graded Module Structure and Stabilization

In arXiv:1308.3152, the author proved that the Khovanov-Rozansky homology $\mathcal{H}_N$ with potential $ax^{N+1}$ is an invariant for transverse links in the standard contact $3$-sphere. In the current paper, we study the $\mathbb{Z}_2 \oplus \mathbb{Z}^{\oplus 3}$-graded $\mathbb{Q}[a]$-module structure of $\mathcal{H}_N$, which leads to better understanding of the effect of stabilization on $\mathcal{H}_N$. As an application, we compute $\mathcal{H}_N$ for all transverse unknots.

Khovanov 辫结拉斯尼亚模量，输入为 $1$ 维

Khovanov skein lasagna modules with $1$-dimensional inputs

本文构造了以 Rozansky-Willis 同调为输入的 Khovanov 辫结拉斯尼亚模量变体，证明其函子性，并给出微分同胚群、Gluck 扭转及 $\mathfrak{gl}_2$ 泡沫的新结果。

Khovanov-Rozansky Homology and Topological Strings

We conjecture a relation between the sl(N) knot homology, recently introduced by Khovanov and Rozansky, and the spectrum of BPS states captured by open topological strings. This conjecture leads to new regularities among the sl(N) knot homology groups and suggests that they can be interpreted directly in topological string theory. We use this approach in various examples to predict the sl(N) knot homology groups for all values of N. We verify that our predictions pass some non-trivial checks.

环形 Khovanov--Rozansky 同调的评估

Evaluations of annular Khovanov--Rozansky homology

本文用自由对称幺半范畴的同伦范畴描述环形 Khovanov-Rozansky 同调函子的万有目标，并借助 plethystic 变换的范畴类比刻画 Coxeter 辫不变量，同时证明缆结缠结上的对称群作用并构造计算广义 Hopf 链环同调的谱序列。

Introduction to Khovanov Homologies. III. A new and simple tensor-algebra construction of Khovanov-Rozansky invariants

We continue to develop the tensor-algebra approach to knot polynomials with the goal to present the story in elementary and comprehensible form. The previously reviewed description of Khovanov cohomologies for the gauge group of rank N-1=1 was based on the cut-and-join calculus of the planar cycles, which are involved rather artificially. We substitute them by alternative and natural set of cycles, not obligatory planar. Then the whole construction is straightforwardly lifted from SL(2) to SL(N) and reproduces Khovanov-Rozansky (KR) polynomials, simultaneously for all values of N. No matrix fa…

Khovanov-Rozansky homology and higher Catalan sequences

We give a simple recursion which computes the triply graded Khovanov-Rozansky homology of several infinite families of knots and links, including the $(n,nm\pm 1)$ and $(n,nm)$ torus links for $n,m\geq 1$. We interpret our results in terms of Catalan combinatorics, proving a conjecture of Gorsky's. Our computations agree with predictions coming from Hilbert schemes and rational DAHA, which also proves the Gorsky-Oblomkov-Rasmussen-Shende conjectures in these cases. Additionally, our results suggest a topological interpretation of the symmetric functions which appear in the context of the $m$-s…

Hilbert schemes and $y$-ification of Khovanov-Rozansky homology

We define a deformation of the triply graded Khovanov-Rozansky homology of a link $L$ depending on a choice of parameters $y_c$ for each component of $L$, which satisfies link-splitting properties similar to the Batson-Seed invariant. Keeping the $y_c$ as formal variables yields a link homology valued in triply graded modules over $\mathbb{Q}[x_c,y_c]_{c\in π_0(L)}$. We conjecture that this invariant restores the missing $Q\leftrightarrow TQ^{-1}$ symmetry of the triply graded Khovanov-Rozansky homology, and in addition satisfies a number of predictions coming from a conjectural connection wit…

二分链环的 Khovanov-Rozansky 循环演算

Khovanov-Rozansky cycle calculus for bipartite links

本文提出二分演算，将 Kauffman 平面展开推广到任意 N，在二分链环上以 3^n 维超立方体直接构造映射、微分和复形，从而简化 Khovanov-Rozansky 演算并重现其多项式。

Knot Floer Homology and Khovanov-Rozansky Homology for Singular Links

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 …

Coxeter结的Khovanov-Rozansky同调与任意直线下路径的Schröder多项式

Khovanov-Rozansky homology of Coxeter knots and Schröder polynomials for paths under any line

本文引入由三角划分索引的广义Schröder多项式，证明其与Coxeter结的Khovanov-Rozansky同调Poincaré级数一致，并借此证明了ORS猜想的特化及$(q,t)$-Schröder定理的三角版本。

Monodromic model for Khovanov-Rozansky homology

We describe a new geometric model for the Hochschild cohomology of Soergel bimodules based on the monodromic Hecke category studied earlier by the first author and Yun. Moreover, we identify the objects representing individual Hochschild cohomology groups (for the zero and the top degree cohomology this reduces to an earlier result of Gorsky, Hogancamp, Mellit and Nakagane). These objects turn out to be closely related to explicit character sheaves corresponding to exterior powers of the reflection representation of the Weyl group. Applying the described functors to the images of braids in the…

$\mathfrak{gl}_n$-webs, 范畴化与 Khovanov-Rozansky 同调

$\mathfrak{gl}_n$-webs, categorification and Khovanov-Rozansky homologies

本文利用范畴化 q-skew Howe 对偶性，为 $\mathfrak{gl}_n$-web 代数 $H_n(\vec{k})$ 构造显式基，并应用于建立厚分圆 KLR 代数的幂等截断与 $H_n(\vec{k})$ 之间的同构，以及 $\mathfrak{gl}_n$-web 的 2-同态空间的显式分次细胞基，从而给出可计算的着色 Khovanov-Rozansky $\mathfrak{gl}_n$-链环同调。

Sl(N) link homology using foams and the Kapustin-Li formula

We use foams to give a topological construction of a rational link homology categorifying the slN link invariant, for N>3. To evaluate closed foams we use the Kapustin-Li formula adapted to foams by Khovanov and Rozansky. We show that for any link our homology is isomorphic to Khovanov and Rozansky's.

Are Khovanov-Rozansky polynomials consistent with evolution in the space of knots?

$R$-coloured knot polynomials for $m$-strand torus knots $Torus_{[m,n]}$ are described by the Rosso-Jones formula, which is an example of evolution in $n$ with Lyapunov exponents, labelled by Young diagrams from $R^{\otimes m}$. This means that they satisfy a finite-difference equation (recursion) of finite degree. For the gauge group $SL(N)$ only diagrams with no more than $N$ lines can contribute and the recursion degree is reduced. We claim that these properties (evolution/recursion and reduction) persist for Khovanov-Rozansky (KR) polynomials, obtained by additional factorization modulo $1…

Mutation invariance of Khovanov homology over $\mathbb{F}_2$

We prove that Khovanov homology and Lee homology with coefficients in $\mathbb{F}_2$ are invariant under component-preserving link mutations.

A spectral sequence for Khovanov homology with an application to (3,q)-torus links

A spectral sequence converging to Khovanov homology is constructed which is applied to calculate the rational Khovanov homology of (3,q)-torus links.

Bar-Natan's Khovanov homology for coloured links

Using Bar-Natan's Khovanov homology we define a homology theory for coloured, oriented, framed links. We then compute this explicitly.

Does Khovanov homology detect the unknot?

We determine a wide class of knots, which includes unknotting number one knots, within which Khovanov homology detects the unknot. A corollary is that the Khovanov homology of many satellite knots, including the Whitehead double, detects the unknot.

Khovanov homology of the 2-cable detects the unknot

We prove that the Khovanov homology of the 2-cable detects the unknot. A corollary is that Khovanov's categorification of the 2-colored Jones polynomial detects the unknot.