Rostislav Akhmechet
We construct an equivariant version of annular Khovanov homology via the Frobenius algebra associated with $U(1) \times U(1)$-equivariant cohomology of $\mathbb{CP}^1$. Motivated by the relationship between the Temperley-Lieb algebra and annular Khovanov homology, we also introduce an equivariant analogue of the Temperley-Lieb algebra.
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We define a third grading on Khovanov homology, which is an invariant of annular links but changes by $\pm 1$ under stabilization. We illustrate the use of our computer implementation, and give some example calculations.
Let $\widetilde{K}$ be a two-periodic braid and let $K$ be its quotient. In this paper we show there is a spectral sequence from the next-to-top winding number grading of the sutured annular Khovanov homology of the closure of $\widetilde{K}$ to the next-to-top winding number grading of the sutured annular Khovanov homology of the closure of $K$.
We exhibit infinite families of annular links for which the maximum non-zero annular Khovanov grading grows infinitely large but the maximum non-zero annular Floer-theoretic gradings are bounded. We also show this phenomenon exists at the decategorified level for some of the infinite families. Our computations provide further evidence for the wrapping conjecture of Hoste-Przytycki and its categorified analogue. Additionally, we show that certain satellite operations cannot be used to construct counterexamples to the categorified wrapping conjecture. We also extend the Batson-Seed link splittin…
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。
We define an annular version of odd Khovanov homology and prove that it carries an action of the Lie superalgebra $\mathfrak{gl}(1|1)$ which is preserved under annular Reidemeister moves.
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…
Let L be a link in a thickened annulus. We show that its sutured annular Khovanov homology carries an action of the exterior current algebra of the Lie algebra sl_2. When L is an m-framed n-cable of a knot K in the three-sphere, its sutured annular Khovanov homology carries a commuting action of the symmetric group S_n. One therefore obtains a "knotted" Schur-Weyl representation that agrees with classical sl_2 Schur-Weyl duality when K is the Seifert-framed unknot.
We prove that the Khovanov-Lee complex of an oriented link, L, in a thickened annulus, A x I, has the structure of a bifiltered complex whose filtered chain homotopy type is an invariant of the isotopy class of L in A x I. Using ideas of Ozsvath-Stipsicz-Szabo as reinterpreted by Livingston, we use this structure to define a family of annular Rasmussen invariants that yield information about annular and non-annular cobordisms. Focusing on the special case of annular links obtained as braid closures, we use the behavior of the annular Rasmussen invariants to obtain a necessary condition for bra…
Quantum topological Hochschild homology and annular Khovanov spectra
本文提出量子拓扑Hochschild同调（qTHH）并构建其稳定同调细化，用于量子环状Khovanov同调的构造，证明其与现有理论一致。
Let $L$ be a link in a thickened annulus. Grigsby-Licata-Wehrli showed that the annular Khovanov homology of $L$ is equipped with an action of $sl_2(\wedge)$, the exterior current algebra of the Lie algebra $sl_2$. In this paper, we upgrade this result to the setting of $L_\infty$-algebras and modules. That is, we show that $sl_2(\wedge)$ is an $L_\infty$-algebra and that the annular Khovanov homology of $L$ is an $L_\infty$-module over $sl_2(\wedge)$. Up to $L_\infty$-quasi-isomorphism, this structure is invariant under Reidemeister moves. Finally, we include explicit formulas to compute the …
For a 2-periodic link $\tilde L$ in the thickened annulus and its quotient link $L$, we exhibit a spectral sequence with $E^1 \cong AKh(\tilde L) \otimes_{\mathbb{F}_2} \mathbb{F}_2[θ, θ^{-1}] \rightrightarrows E^\infty \cong AKh(L) \otimes_{\mathbb{F}_2} \mathbb{F}_2[θ, θ^{-1}].$ This spectral sequence splits along quantum and $sl_2$ weight space gradings, proving a rank inequality $rank\ AKh^{j,k}(L) \leq rank\ AKh^{2j-k,k} (\tilde L)$ for every pair of quantum and $sl_2$ weight space gradings $(j,k)$. We also present a few decategorified consequences and discuss partial results toward a sim…
We study Khovanov homology over the Frobenius algebra $\mathbb{F}[U,V,X]/((X-U)(X-V))$, or $U(1) \times U(1)$-equivariant Khovanov homology, and extract two families of concordance invariants using the algebraic $U$-power and $V$-power filtrations on the chain complex. We also further develop the reduced version of the theory and study its behavior under mirroring.
Symmetries of equivariant Khovanov homology
本文研究等价Khovanov同调的对称性，提出反置和积分提升操作，并将其与Rasmussen $s$-不变量联系起来。
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.
Lawrence Roberts, extending the work of Ozsvath-Szabo, showed how to associate to a link, L, in the complement of a fixed unknot, B, in S^3, a spectral sequence from the Khovanov homology of a link in a thickened annulus to the knot Floer homology of the preimage of B inside the double-branched cover of L. In a previous paper, we extended Ozsvath-Szabo's spectral sequence in a different direction, constructing for each knot K in S^3 and each positive integer n, a spectral sequence from Khovanov's categorification of the reduced, n-colored Jones polynomial to the sutured Floer homology of a red…
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。
Beliakova-Putyra-Wehrli studied various kinds of traces, in relation to annular Khovanov homology. In particular, to a graded algebra and a graded bimodule over it, they associate a quantum Hochschild homology of the algebra with coefficients in the bimodule, and use this to obtain a deformation of the annular Khovanov homology of a link. A spectral refinement of the resulting invariant was recently given by Akhmechet-Krushkal-Willis. In this short note we observe that quantum Hochschild homology is a composition of two familiar operations, and give a short proof that it gives an invariant of …
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…
In 2001, Khovanov and Seidel constructed a faithful action of the (m+1)-strand braid group on the derived category of left modules over a quiver algebra, A_m. We interpret the Hochschild homology of the Khovanov-Seidel braid invariant as a direct summand of the sutured Khovanov homology of the annular braid closure.
For every positive integer $n$ we construct a bigraded homology theory for links, such that the corresponding invariant of the unknot is closely related to the U(n)-equivariant cohomology ring of $\mathbb{CP}^{n-1}$; our construction specializes to the Khovanov-Rozansky $sl_n$-homology. We are motivated by the "universal" rank two Frobenius extension studied by M. Khovanov in \cite{Kh3} for $sl_2$-homology.
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。
We construct a Kirby color in the setting of Khovanov homology: an ind-object of the annular Bar-Natan category that is equipped with a natural handle slide isomorphism. Using functoriality and cabling properties of Khovanov homology, we define a Kirby-colored Khovanov homology that is invariant under the handle slide Kirby move, up to isomorphism. Via the Manolescu--Neithalath 2-handle formula, Kirby-colored Khovanov homology agrees with the $\mathfrak{gl}_{2}$ skein lasagna module, hence is an invariant of $4$-dimensional $2$-handlebodies.
On unification of colored annular sl(2) knot homology
本研究在量子环状Bar-Natan范畴下证明平凡纽结的三种着色sl(2)同调理论等价，通过切比雪夫系统唯一性完成证明，还验证了两类投影算子类的一致性，计算了Khovanov弧代数的量子Hochschild同调，并提出推广至所有纽结的猜想。
We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. For each $r\geq 2$ we associate to an annular link $L$ a naive $\mathbb{Z}/r\mathbb{Z}$-equivariant spectrum whose cohomology is isomorphic to the quantum annular homology of $L$ as modules over $\mathbb{Z}[\mathbb{Z}/r\mathbb{Z}]$. The construction relies on an equivariant version of the Burnside category approach of Lawson, Lipshitz and Sarkar. The quotient under the cyclic group action is shown to recover the stable homotopy refinement of annular Khovanov…
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.
We introduce two monoidal supercategories: the odd dotted Temperley-Lieb category $\mathcal{T\!L}_{o,\bullet}(δ)$, which is a generalization of the odd Temperley-Lieb category studied by Brundan and Ellis, and the odd annular Bar-Natan category $\mathcal{BN}_{\!o}(\mathbb{A})$, which generalizes the odd Bar-Natan category studied by Putyra. We then show there is an equivalence of categories between them if $δ=0$. We use this equivalence to better understand the action of the Lie superalgebra $\mathfrak{gl}(1|1)$ on the odd Khovanov homology of a knot in a thickened annulus found by Grigsby and…
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。
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。
Anchored foams and annular homology
该研究通过泡沫赋值方法，定义实心环面中链环的等变SL(2)、SL(3)同调，引入锚点概念修正泡沫赋值规则，同时覆盖有向与无向SL(3)泡沫情形。
Annular SL(2) and SL(3) web algebras
本文利用环带泡沫拓扑量子场论定义环带上的等变SL(2)与SL(3)编织代数，并建立缠结不变量及非椭圆编织与权格闭合路径的双射对应。
Cosmetic operations and Khovanov multicurves
本文结合 Hanselman 的结果与 Khovanov 多重曲线不变量，证明强可逆纽结的等变美容手术猜想，并重新证明美容交叉猜想相关结果及 Conway 缠结分裂性的可检测性。
A $y$-ification of Khovanov homology
本文通过构造Khovanov同调的$y$-化及其等变版本，定义了$\mathfrak{sl}_2$作用，并证明其与HOMFLY-PT同调到Khovanov同调的谱序列相容，从而区分具有相同同调的结对。
An Intrinsic $L_{\infty}$-Algebra on the Khovanov-Sano Complex
本文利用BV形式体系与同伦转移定理，将Khovanov-Sano复形同调上的非平凡$L_{\infty}$结构提升至链层面，得到典范链环不变量。
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。
This thesis studies the Bar-Natan skein module of the solid torus with a particular boundary curve system, and in particular a diagrammatic presentation of it due to Russell. This module has deep connections to topology and categorification: it is isomorphic to both the total homology of the (n,n)-Springer variety and the 0th Hochschild homology of the Khovanov arc ring H^n. We can also view the Bar-Natan--Russell skein module from a representation-theoretic viewpoint as an extension of the Frenkel--Khovanov graphical description of the Lusztig dual canonical basis of the nth tensor power of t…
Motivated by topology, we develop a general theory of traces and shadows for an endobicategory, which is a~pair: bicategory $\mathbf{C}$ and endobifunctor $Σ\colon \mathbf C \to\mathbf C$. For a graded linear bicategory and a fixed invertible parameter $q$, we quantize this theory by using the endofunctor $Σ_q$ such that $Σ_q α:=q^{-°α}Σα$ for any 2-morphism $α$ and coincides with $Σ$ otherwise. Applying the quantized trace to the~bicategory of Chen-Khovanov bimodules we get a new triply graded link homology theory called quantum annular link homology. If $q=1$ we reproduce Asaeda-Przytycki-Si…
On Diagrammatic Categorification of Verma Modules I: Braiding
本文通过图解双模的导出张量积范畴化 Verma 模的 R-矩阵，构造辫子函子并证明辫群作用，为纽结补空间建立 Khovanov 同调，并与环形 Khovanov 同调比较。
Using derived categories of equivariant coherent sheaves, we construct a categorification of the tangle calculus associated to sl(2) and its standard representation. Our construction is related to that of Seidel-Smith by homological mirror symmetry. We show that the resulting doubly graded knot homology agrees with Khovanov homology.
Using derived categories of equivariant coherent sheaves we construct a knot homology theory which categorifies the quantum sl(m) knot polynomial. Our knot homology naturally satisfies the categorified MOY relations and is conjecturally isomorphic to Khovanov-Rozansky homology. Our construction is motivated by the geometric Satake correspondence and is related to Manolescu's by homological mirror symmetry.
Let $\mathcal{U}$ be the unipotent variety of a complex reductive group $G$. Fix opposed Borel subgroups $B_\pm \subseteq G$ with unipotent radicals $U_\pm$. The map that sends $x_+x_- \mapsto x_+x_-x_+^{-1}$ for all $x_\pm \in U_\pm$ restricts to a map from $U_+U_- \cap gB_+$ into $\mathcal{U} \cap gB_+$, for any $g$. We conjecture that the restricted map forms half of a homotopy equivalence between these varieties, and thus, induces a weight-preserving isomorphism between their compactly-supported cohomologies. Noting that the map is equivariant with respect to certain actions of $B_+ \cap g…
Symmetry Breaking and Link Homologies II
本文构造过滤的U(r)-等变稳定同伦谱，证明其谱序列E_1项同构于Soergel双模的Hochschild同调，从而计算三度链环同调，并通过扭曲的Borel等变上同调将普适幂级数特化得到sl(n)-链环同调。
From Cherednik algebras to knot homology via cuspidal D-modules
本文证明环面纽结的三重分级 Khovanov-Rozansky 同调可由有理切列德尼克代数的有限维表示及其尖点特征 D-模的 Hodge 滤过恢复，并通过 shuffle 代数与 Hilbert 概型等变 K 理论的关系证明了有理主猜想。
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