sl(3) link homology

Annular Evaluation and Link Homology

We use categorical annular evaluation to give a uniform construction of both $\mathfrak{sl}_n$ and HOMFLYPT Khovanov-Rozansky link homology, as well as annular versions of these theories. Variations on our construction yield $\mathfrak{gl}_{-n}$ link homology, i.e. a link homology theory associated to the Lie superalgebra $\mathfrak{gl}_{0|n}$, both for links in $S^3$ and in the thickened annulus. In the $n=2$ case, this produces a categorification of the Jones polynomial that we show is distinct from Khovanov homology, and gives a finite-dimensional categorification of the colored Jones polyn…

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…

A link surgery spectral sequence in monopole Floer homology

To a link L in the 3-sphere, we associate a spectral sequence whose E^2 page is the reduced Khovanov homology of L and which converges to a version of the monopole Floer homology of the branched double cover. The pages E^k for k > 1 depend only on the mutation equivalence class of L. We define a mod 2 grading on the spectral sequence which interpolates between the delta-grading on Khovanov homology and the mod 2 grading on Floer homology. We also derive a new formula for link signature that is well-adapted to Khovanov homology. More generally, we construct new bigraded invariants of a framed l…

The 1,2-coloured HOMFLY-PT link homology

In this paper we define the 1,2-coloured HOMFLY-PT link homology and prove that it is a link invariant. We conjecture that this homology categorifies the coloured HOMFLY-PT polynomial for links whose components are labelled 1 or 2.

三分次链环同调中的最后三个T次

The Last Three T-degrees in Triply-Graded Link Homology

本研究采用Soergel双模Hochschild上同调的图示方法，显式计算了正/负辫链闭包链环的约化三分次链环同调在最高/最低三个T次中的R-模结构，发现同调常为零且具显著一致性。

Link homology and categorification

This is a short survey of algebro-combinatorial link homology theories which have the Jones polynomial and other link polynomials as their Euler characteristics.

Positive half of the Witt algebra acts on triply graded link homology

The positive half of the Witt algebra is the Lie algebra spanned by vector fields x^{m+1} d/dx acting as differentiations on the polynomial algebra Q[x] upon which the Soergel bimodule construction of triply graded link homology is based. We show that this action of Witt algebra can be extended to the link homology.

Notes on link homology

This article consists of six lectures on the categorification of the Burau representation and on link homology groups which categorify the Jones and the HOMFLY-PT polynomial. The notes are based on the lecture course at the PCMI 2006 summer school in Park City, Utah.

Homology of torus links

In this paper we show that there is a cut-off in the Khovanov homology of $(2k,2kn)$-torus links, namely that the maximal homological degree of non-zero homology group of $(2k,2kn)$-torus link is $2k^2n$. Furthermore, we calculate explicitely the homology groups in homological degree $2k^2n$ and prove that it coincides with the centre of the ring $H^k$ of crossingless matchings, introduced by M. Khovanov in \cite{tan}. Also we give an explicit formula for the ranks of the homology groups of $(3,n)$-torus knots for every $n\in\N$.

Triply-graded link homology and Hochschild homology of Soergel bimodules

We trade matrix factorizations and Koszul complexes for Hochschild homology of Soergel bimodules to modify the construction of triply-graded link homology and relate it to Kazhdan-Lusztig theory.

Knot homology via derived categories of coherent sheaves IV, coloured links

We define a deformation of our earlier link homologies for fundamental representations of sl_m. The deformed homology of a link is isomorphic to the deformed homology of the disjoint union of its components. Moreover, there exists a spectral sequence starting with the old homology and converging to this deformed homology.

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.

Categorifications of the colored Jones polynomial

The colored Jones polynomial of links has two natural normalizations: one in which the n-colored unknot evaluates to [n+1], the quantum dimension of the (n+1)-dimensional irreducible representation of quantum sl(2), and the other in which it evaluates to 1. For each normalization we construct a bigraded cohomology theory of links with the colored Jones polynomial as the Euler characteristic.

Matrix factorizations and link homology

For each positive integer n the HOMFLY polynomial of links specializes to a one-variable polynomial that can be recovered from the representation theory of quantum sl(n). For each such n we build a doubly-graded homology theory of links with this polynomial as the Euler characteristic. The core of our construction utilizes the theory of matrix factorizations, which provide a linear algebra description of maximal Cohen-Macaulay modules on isolated hypersurface singularities.

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.

On sl(N) link homology with mod N coefficients

We construct an operator on sl(N) link homology with coefficients in a ring whose characteristic divides N. When P is prime, we use this operator to exhibit structural features of sl(P) link homology that are special to characteristic P that generalize well-known features of Khovanov homology that are special to characteristic 2.

Link homology and Frobenius extensions

We explain how rank two Frobenius extensions of commutative rings lead to link homology theories and discuss relations between these theories, Bar-Natan theories, equivariant cohomology and the Rasmussen invariant.

Integral HOMFLY-PT and sl(n)-link homology

Using the diagrammatic calculus for Soergel bimodules, developed by B. Elias and M. Khovanov, as well as Rasmussen's spectral sequence, we construct an integral version of HOMFLY-PT and sl(n)-link homology.

An elementary construction of Khovanov-Rozansky type link homology

In this article, we give an elementary construction of homological invariants of links presented by braid closures. The Euler characteristic of this complex is equal to quantum polynomial invariant of link.

Equivariant colored sl(N)-homology for links

We construct an equivariant colored sl(N)-homology for links, which generalizes both the colored sl(N)-homology defined by the author and the equivariant sl(N)-homology defined by Krasner. The construction is a straightforward generalization of that of the colored sl(N)-homology. The proof of invariance is based on a simple observation which allows us to translate the proof of the invariance of the colored sl(N)-homology into the new setting. As an application, we prove that deformations over C of the colored sl(N)-homology are link invariants. We also construct a spectral sequence connecting …

The Universal sl_2 Link Homology Theory

We explore the complex associated to a link in the geometric formalism of Khovanov's (n=2) link homology theory, determine its exact underlying algebraic structure and find its precise universality properties for link homology functors. We present new methods of extracting all known link homology theories directly from this universal complex, and determine its relative strength as a link invariant by specifying the amount of information held within the complex. We achieve these goals by finding a complex isomorphism which reduces the complex into one in a simpler category. We introduce few too…

A refinement of Khovanov-Rozansky link homology

This paper has been withdrawn by the author due to an error in the proof of Theorem 1.

Khovanov-Rozansky homology of two-bridge knots and links

We compute the reduced version of Khovanov and Rozansky's sl(N) homology for two-bridge knots and links. The answer is expressed in terms of the HOMFLY polynomial and signature.

Khovanov homology of links and graphs

In this thesis we work with Khovanov homology of links and its generalizations, as well as with the homology of graphs. Khovanov homology of links consists of graded chain complexes which are link invariants, up to chain homotopy, with graded Euler characteristic equal to the Jones polynomial of the link. Hence, it can be regarded as the "categorif\mbox{}ication" of the Jones polynomial. \indent We prove that the f\mbox{}irst homology group of positive braid knots is trivial. Futhermore, we prove that non-alternating torus knots are homologically thick. In addition, we show that we can decreas…

有色$\boldsymbol{\frak{gl}_N}$纽结同调的图解分次界

A diagrammatic grading bound for colored $\mathfrak{gl}_N$ link homology

该研究建立了可从任意纽结图读出的有色$\boldsymbol{\frak{gl}_N}$纽结同调的分次界，并将其应用于接触几何，得到Legendrian纽结的Thurston–Bennequin数的界。

Unoriented topological quantum field theory and link homology

We investigate link homology theories for stable equivalence classes of link diagrams on orientable surfaces. We apply (1+1)-dimensional unoriented topological quantum field theories to Bar-Natan's geometric formalism to define new theories for stable equivalence classes.

Khovanov homology for links in thickened multipunctured disks

We define a variant of Khovanov homology for links in thickened disks with multiple punctures. This theory is distinct from the one previously defined by Asaeda, Przytycki, and Sikora, but is related to it by a spectral sequence. Additionally, we show that there are spectral sequences induced by embeddings of thickened surfaces, which recover the spectral sequence from annular Khovanov homology to Khovanov homology as a special case.

在T(2,k) torus链接的链接同调上的一种$\mathfrak{sl}_2$作用

An $\mathfrak{sl}_2$ action on link homology of T(2,k) torus links

本文基于arXiv:2306.10729框架，确定了T(2,k) torus链接的等价Khovanov-Rozansky同调的$\mathfrak{sl}_2$模结构。

A polynomial action on colored sl(2) link homology

We construct an action of a polynomial ring on the colored sl(2) link homology of Cooper-Krushkal, over which this homology is finitely generated. We define a new, related link homology which is finite dimensional, extends to tangles, and categorifies a scalar-multiple of the sl(2) Reshetikhin-Turaev invariant. We expect this homology to be functorial under 4-dimensional cobordisms. The polynomial action is related to a conjecture of Gorsky-Oblomkov-Rasmussen-Shende on the stable Khovanov homology of torus knots, and as an application we obtain a weak version of this conjecture. A key new ingr…

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.

从光滑4流形边界的链接同调论中表面不变量

Invariants of surfaces in smooth 4-manifolds from link homology

本文提出了一种基于变形$\mathfrak{gl}_N$链接同调的skein lasagna模块，用于构建光滑4流形边界的链接不变量，并建立了其分解定理和泛函性扩展方法。

HOMFLY-PT homology for general link diagrams and braidlike isotopy

Khovanov and Rozansky's categorification of the HOMFLY-PT polynomial is invariant under braidlike isotopies for any link diagram and Markov moves for braid closures. To define HOMFLY-PT homology, they required a link to be presented as a braid closure, because they did not prove invariance under the other oriented Reidemeister moves. In this text we prove that the Reidemeister IIb move fails in HOMFLY-PT homology by using virtual crossing filtrations of the author and Rozansky. The decategorification of HOMFLY-PT homology for general link diagrams gives a deformed version of the HOMFLY-PT poly…

A Link Splitting Spectral Sequence in Khovanov Homology

We construct a new spectral sequence beginning at the Khovanov homology of a link and converging to the Khovanov homology of the disjoint union of its components. The page at which the sequence collapses gives a lower bound on the splitting number of the link, the minimum number of times its components must be passed through one another in order to completely separate them. In addition, we build on work of Kronheimer-Mrowka and Hedden-Ni to show that Khovanov homology detects the unlink.

4股线上Coxeter辫子的三重复分次链环同调

Triply Graded Link Homology for Coxeter Braids on 4 Strands

本文计算了4股线上Coxeter辫子的三重复分次Khovanov-Rozansky同调，为相关链环同调研究提供了具体计算结果。

通过A型Soergel双模构造带柄体中链环的HOMFLYPT同调

HOMFLYPT homology for links in handlebodies via type A Soergel bimodules

本文利用经典辫群子群与奇异Soergel双模的范畴作用，将3-球中着色HOMFLYPT同调推广为亏格g带柄体中链环的三重分次不变量。

Categorified Young symmetrizers and stable homology of torus links

We show that the triply graded Khovanov-Rozansky homology of the torus link $T_{n,k}$ stablizes as $k\to \infty$. We explicitly compute the stable homology (as a ring), which proves a conjecture of Gorsky-Oblomkov-Rasmussen-Shende. To accomplish this, we construct complexes $P_n$ of Soergel bimodules which categorify the Young symmetrizers corresponding to one-row partitions and show that $P_n$ is a stable limit of Rouquier complexes. A certain derived endomorphism ring of $P_n$ computes the aforementioned stable homology of torus links.

On Link Homology Theories from Extended Cobordisms

This paper is devoted to the study of algebraic structures leading to link homology theories. The originally used structures of Frobenius algebra and/or TQFT are modified in two directions. First, we refine 2-dimensional cobordisms by taking into account their embedding into the three space. Secondly, we extend the underlying cobordism category to a 2-category, where the usual relations hold up to 2-isomorphisms. The corresponding abelian 2-functor is called an extended quantum field theory (EQFT). We show that the Khovanov homology, the nested Khovanov homology, extracted by Stroppel and Webs…

Link homology and Frobenius extensions II

The first two sections of the paper provide a convenient scheme and additional diagrammatics for working with Frobenius extensions responsible for key flavors of equivariant SL(2) link homology theories. The goal is to clarify some basic structures in the theory and propose a setup to work over sufficiently non-degenerate base rings. The third section works out two related SL(2) evaluations for seamed surfaces.

Categorified Young symmetrizers and stable homology of torus links II

We construct complexes $P_{1^n}$ of Soergel bimodules which categorify the Young idempotents corresponding to one-column partitions. A beautiful recent conjecture of Gorsky-Rasmussen relates the Hochschild homology of categorified Young idempotents with the flag Hilbert scheme. We prove this conjecture for $P_{1^n}$ and its twisted variants. We also show that this homology is also a certain limit of Khovanov-Rozansky homologies of torus links. Along the way we obtain several combinatorial results which could be of independent interest.

N-Degeneracy in rack homology and link invariants

The aim of this paper is to define a homology theory for racks with finite rank N and use it to define invariants of knots generalizing the CJKLS 2-cocycle invariants related to the invariants defined in [15]. For this purpose, we prove that N -degenerate chains form a sub-complex of the classical complex defining rack homology. If a rack has rack rank N = 1 then it is a quandle and our homology theory coincides with the CKJLS homology theory [6]. Nontrivial cocycles are used to define invariants of knots and examples of calculations for classical knots with up to 8 crossings and classical lin…

An invariant of link cobordisms from Khovanov homology

In [Duke Math. J. 101 (1999) 359-426], Mikhail Khovanov constructed a homology theory for oriented links, whose graded Euler characteristic is the Jones polynomial. He also explained how every link cobordism between two links induces a homomorphism between their homology groups, and he conjectured the invariance (up to sign) of this homomorphism under ambient isotopy of the link cobordism. In this paper we prove this conjecture, after having made a necessary improvement on its statement. We also introduce polynomial Lefschetz numbers of cobordisms from a link to itself such that the Lefschetz …

半充分链环的Khovanov同调中的挠

Torsion in Khovanov homology of semi-adequate links

针对Khovanov链环同调中$\text{Z}_2$-挠存在性的Shumakovitch猜想，研究者借助图的色上同调分析半充分链环的Khovanov同调挠，得到显式公式并证明对应Tait型图有长度≥3的环时存在$\text{Z}_2$-挠，还提出挠与辫指数相关的猜想。

SO(3) Homology of Graphs and Links

The SO(3) Kauffman polynomial and the chromatic polynomial of planar graphs are categorified by a unique extension of the Khovanov homology framework. Many structural observations and computations of homologies of knots and spin networks are included.

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.