Skein theory for the ADE planar algebras
Stephen Bigelow
给出了$ADE$子因子对应的平面代数的生成元和关系。我们还给出了将任意图表示为这些基图的线性组合的基和算法。
We give generators and relations for the planar algebras corresponding to $ADE$ subfactors. We also give a basis and an algorithm to express an arbitrary diagram as a linear combination of these basis diagrams.
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We describe an explicit finite presentation for a finite depth subfactor planar algebra. We also show that such planar algebras are singly generated with the generator subject to finitely many relations.
We give a combinatorial description of the ``$D_{2n}$ planar algebra,'' by generators and relations. We explain how the generator interacts with the Temperley-Lieb braiding. This shows the previously known braiding on the even part extends to a `braiding up to sign' on the entire planar algebra. We give a direct proof that our relations are consistent (using this `braiding up to sign'), give a complete description of the associated tensor category and principal graph, and show that the planar algebra is positive definite. These facts allow us to identify our combinatorial construction with the…
Skein Theory for Affine A Subfactor Planar Algebras
本研究推进Kuperberg纲领，给出指标为4的仿射A子因子平面代数的生成元与关系表示，以纯图解方式新证这类平面代数的存在数量及对应范畴的幺半等价性，提供全新图解方法。
Presentations of Kauffman bracket skein algebras of planar surfaces
本文在含特定可逆元的交换环上，通过显式生成元与关系展示了平面曲面的Kauffman括号skein代数，该展示独立于环且可视为幺模矩阵迹代数的量子化。
We construct a new subfactor planar algebra, and as a corollary a new subfactor, with the `extended Haagerup' principal graph pair. This completes the classification of irreducible amenable subfactors with index in the range $(4,3+\sqrt{3})$, which was initiated by Haagerup in 1993. We prove that the subfactor planar algebra with these principal graphs is unique. We give a skein theoretic description, and a description as a subalgebra generated by a certain element in the graph planar algebra of its principal graph. In the skein theoretic description there is an explicit algorithm for evaluati…
Bisch and Jones suggested the skein theoretic classification of planar algebras and investigated the ones generated by 2-boxes with the second author. In this paper, we consider 3-box generators and classify subfactor planar algebras generated by a non-trivial 3-box satisfying a relation proposed by Thurston. The subfactor planar algebras in the classification are either $E^6$ or the ones from representations of quantum $SU(N)$. We introduce a new method to determine positivity of planar algebras and new techniques to reduce the complexity of computations.
On skein algebras of planar surfaces
研究平面曲面的 skein 代数结构，通过生成元和关系的理想定义，揭示了不同条件下生成元数量及关系的度数限制，推动了 Kirby 问题列表中 Problem 1.92 (J) 的解决。
The main purpose of this paper is to classify exchange relation planar algebras with 4 dimensional 2-boxes. Besides its skein theory, we emphasize the positivity of subfactor planar algebras based on the Schur product theorem. We will discuss the lattice of projections of 2-boxes, specifically the rank of the projections. From this point, several results about biprojections are obtained. The key break of the classification is to show the existence of a biprojection. By this method, we also classify another two families of subfactor planar algebras, subfactor planar algebras generated by 2-boxe…
A polynomial invariant of planar curves
本文通过 skein 关系构造平面曲线的多项式不变量，推广了 Arnold 的数值不变量 St。
We give a new type of Schur-Weyl duality for the representations of a family of quantum subgroups and their centralizer algebra. We define and classify singly-generated, Yang-Baxter relation planar algebras. We present the skein theoretic construction of a new parameterized planar algebra. We construct infinitely many new subfactors and unitary fusion categories, and compute their trace formula as a closed-form expression, in terms of Young diagrams.
Homotopy and q-homotopy skein modules of 3-manifolds: an example in Algebra Situs
代数Situs是数学的一个分支，其根源在于琼斯的建设，他的多项式不变的联系和德林费尔德的工作量子群。它包括理论的量子不变量的节点和3-流形，代数拓扑的基础上结，运算，平面代数，q-变形，量子群，并重叠与代数几何，非交换几何和统计力学。基于结的代数拓扑学可以被描述为通过考虑流形中的链接（子流形）及其代数结构来研究流形的性质。这门学科的主要对象是绞链模，它是流形中链路的环境合痕类上的自由模通过适当选择局部（绞链）关系的等价物。在这堂课中，我们集中讨论三维流形的一个相对简单的例子--q-同伦的skein模。这个绞链模块已经有了理论的许多成分：代数结构，相关的李代数，量子化，状态模型.
Variational derivation of exact skein relations from Chern--Simons theories
在陈--西蒙斯理论中，威尔逊圈的期望值是纽结不变量.它的绞结关系已经以各种方式推导出来，包括变分法，其中对环进行小变形并评估变化。后一种方法只允许获得绞纱关系的近似表达式。我们提出了一个概括的这个想法，允许计算的确切形式的绞关系。此外，它需要推广所得到的结不变量相交结和链接的方式符合曼德尔维恒等式满足的威尔逊圈。这使得第一次推导出完整的表达式结不变量是合适的候选人量子态的重力（和超重力）的循环表示。新的方法导致了一些新的见解，在相交结理论，特别是非平面交叉和交叉扭结的作用。
We consider a deformation of the Robert-Wagner foam evaluation formula, with an eye toward a relation to formal groups. Integrality of the deformed evaluation is established, giving rise to state spaces for planar GL(N) MOY graphs (Murakami-Ohtsuki-Yamada graphs). Skein relations for the deformation are worked out in details in the GL(2) case. These skein relations deform GL(2) foam relations of Beliakova, Hogancamp, Putyra and Wehrli. We establish the Reidemeister move invariance of the resulting chain complexes assigned to link diagrams, giving us a link homology theory.
Generalizing Jones's notion of a planar algebra, we have previously introduced an A_2-planar algebra capturing the structure contained in the double complex pertaining to the subfactor for a finite SU(3) ADE graph with a flat cell system. We now introduce the notion of modules over an A_2-planar algebra, and describe certain irreducible Hilbert A_2-TL-modules. We construct an A_2-graph planar algebra associated to each pair (G,W) given by an SU(3) ADE graph G and a cell system W on G. A partial modular decomposition of these A_2-graph planar algebras is achieved.
We give a diagrammatic presentation of the A_2-Temperley-Lieb algebra. Generalizing Jones' notion of a planar algebra, we formulate an A_2-planar algebra motivated by Kuperberg's A_2-spider. This A_2-planar algebra contains a subfamily of vector spaces which will capture the double complex structure pertaining to the subfactor for a finite SU(3) ADE graph with a flat cell system, including both the periodicity three coming from the A_2-Temperley-Lieb algebra as well as the periodicity two coming from the subfactor basic construction. We use an A_2-planar algebra to obtain a description of the …
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