Marta Asaeda, Charles Frohman
We introduce a new skein module for three manifolds based on properly embedded surfaces and their relations introduced by D.Bar-Natan, and modified by M.Khovanov. We compute the structure of the modules for some manifolds, including Seifert fibred manifolds.
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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…
Using Bar-Natan's Khovanov homology we define a homology theory for coloured, oriented, framed links. We then compute this explicitly.
We investigate Bar-Natan's characteristic two Khovanov link homology theory studying both the filtered and bi-graded theories. The filtered theory is computed explicitly and the bi-graded theory analysed by setting up a family of spectral sequences. The E_2-pages can be described in terms of groups arising from the action of a certain endomorphism on mod 2 Khovanov homology. Some simple consequences are discussed.
Khovanov introduced a cohomology theory for oriented classical links whose graded Euler characteristic is the Jones polynomial. Since Khovanov's theory is functorial for link cobordisms between classical links, we obtain an invariant of a surface-knot, called the {\it Khovanov-Jacobsson number}, by considering the surface-knot as a link cobordism between empty links. In this paper, we define an invariant of a surface-knot which is a generalization of the Khovanov-Jacobsson number by using Bar-Natan's theory, and prove that any $T^2$-knot has the trivial Khovanov-Jacobsson number.
On the $A_\infty$-extension of Bar-Natan multicurves
本文证明了Bar-Natan不变量经$A_\infty$-扩张得到的扭曲复形是缠绕不变量，解决了Kotelskiy等人的猜想，并通过推广的矩阵多因子分解和Koszul形变收缩公式给出证明。
Kirby belts, categorified projectors, and the skein lasagna module of $S^{2}\times{S^{2}}$
本文将 Manolescu-Neithalath 公式解释为同伦余极限，用范畴化投影子计算特定流形的 skein lasagna 模，并证明 $S^2 \times S^2$ 的模平凡，证实 Manolescu 猜想。
A construction of surface skein TQFTs and their extension to 4-dimensional 2-handlebodies
本文提出了一种构造表面 skein TQFT 的方法，并扩展其到 4 维 2-柄体，通过归纳状态和构造和 Kirby 颜色来计算不变量。
On Bar-Natan - van der Veen's perturbed Gaussians
本文研究 Bar-Natan–van der Veen 基于非交换高斯演算提出的多项式时间纽结多项式族，证明其中半数为零，给出三个不变量的具体公式，并分析其在连通和下的行为。
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…
Let $F$ be a finite type surface and $ζ$ a complex root of unity. The Kauffman bracket skein algebra $K_ζ(F)$ is an important object in both classical and quantum topology as it has relations to the character variety, the Teichmüller space, the Jones polynomial, and the Witten-Reshetikhin-Turaev Topological Quantum Field Theories. We compute the rank and trace of $K_ζ(F)$ over its center, and we extend a theorem of Frohman and Kania-Bartoszynska which says the skein algebra has a splitting coming from two pants decompositions of $F$.
We give a geometric interpretation of Bar-Natan's universal invariant for the class of tangles in the 3-ball with four ends: we associate with such 4-ended tangles $T$ multicurves $\widetilde{\operatorname{BN}}(T)$, that is, collections of immersed curves with local systems in the 4-punctured sphere. These multicurves are tangle invariants up to homotopy of the underlying curves and equivalence of the local systems. They satisfy a gluing theorem which recovers the reduced Bar-Natan homology of links in terms of wrapped Lagrangian Floer theory. Furthermore, we use $\widetilde{\operatorname{BN}}…
In this article, we prove the conjecture of Bar-Natan, Garoufalidis, and Khovanov's on the support of the Khovanov's invariants for alternating knots.
Derived skein module
提出面向定向三维流形的导出辫子理论公理框架，以ribbon张量范畴为系数，通过bar构造控制粘合，证明与普通辫子理论的关系，并给出可计算公式和首次计算。
We discuss a new perspective on Khovanov homology, using categorifications of tensor products. While in many ways more technically demanding than Khovanov's approach (and its extension by Bar-Natan), this has distinct advantage of directly connecting Khovanov homology to a categorification of \$(\mathbb{C}^2)^{\otimes \ell}\$, and admitting a direct generalization to other Lie algebras. While the construction discussed is a special case of that given in previous work of the author, this paper contains new results about the special case of \$\mathfrak{sl}_2\$ showing an explicit connection to B…
Azumaya loci of skein algebras
本文计算了闭合曲面在奇数单位根处Kauffman括号绞链代数的Azumaya轨迹，并应用于重新定义Torelli群的射影表示及加强闭合3-流形绞链模商空间维数的已有结果。
The Unicity Theorem and the center of the ${\rm SL}_3$-skein algebra
本文证明了穿孔曲面的 ${\rm SL}_3$-skein 代数在单位根情形下的表示唯一性定理，刻画了其中心生成元，并计算了代数在中心上的秩及一般不可约表示的维数。
Canonical bases of the oriented skein category
本文通过图示方法为定向绞链范畴的每个态射空间构造杠对合与典范基，并在量子化带墙布劳尔代数上得到Kazhdan-Lusztig型基。
Homomorphic expansions are combinatorial invariants of knotted objects, which are universal in the sense that all finite-type (Vassiliev) invariants factor through them. Homomorphic expansions are also important as bridging objects between low-dimensional topology and quantum algebra. For example, homomorphic expansions of parenthesised braids are in one-to-one correspondence with Drinfel'd associators (Bar-Natan 1998), and homomorphic expansions of $w$-foams are in one-to-one correspondence with solutions to the Kashiwara-Vergne (KV) equations (Bar-Natan and the first author, 2017). The sets …
Cohomology theory of links, introduced by the author, is combinatorial. Dror Bar-Natan recently wrote a program that found ranks of cohomology groups of all prime knots with up to 11 crossings. His surprising experimental data is discussed in this note.
On the functoriality of sl(2) tangle homology
本研究构建gl(2)网与泡沫范畴和Bar-Natan的Temperley-Lieb图配边范畴的显式等价，定义弧代数的网版本及拟遗传覆盖以得到严格函子性的链环缠结同调，还给出其与原代数的同构，实现环形链环同调量子化的严格函子性版本。
The algebra $\mathcal{L}_{g,n}(H)$ was introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche and quantizes the character variety of the Riemann surface $Σ_{g,n}\!\setminus\! D$ ($D$ is an open disk). In this article we define a holonomy map in that quantized setting, which associates a tensor with components in $\mathcal{L}_{g,n}(H)$ to tangles in $(Σ_{g,n}\!\setminus\!D) \times [0,1]$, generalizing previous works of Buffenoir-Roche and Bullock-Frohman-Kania-Bartoszynska. We show that holonomy behaves well for the stack product and the action of the mapping class group; then we specializ…
Stated $SL_n$-skein modules, roots of unity, and TQFT
该研究构造了带标架$SL_n$-扭结代数的线性同构，证明了多类分裂映射的单射性，推广了相关扭结空间研究与$SL_2$-TQFT理论，构建了带标架的$SL_n$-TQFT理论。
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