摘要翻译：
本文介绍了一个概念框架，在量子拓扑及其下的代数的背景下，用于分析平面图的色多项式chi(Q)所服从的关系。利用它，我们给出了新的证明，并极大地推广了一些关于色多项式组合的经典结果。特别地，我们证明了Tutte的golden恒等式是SO(N)拓扑量子场论和Birman-Murakami-Wenzl代数的能级对偶的结果。这个恒等式是关于球面任意三角剖分的chi({phi+2}）与同一图的(chi({phi+1}））^2的色多项式的一个显著特征，其中phi表示黄金分割比。本文的新观点解释了Tutte恒等式对于参数q的这些值是特殊的。{em色代数}为分析色多项式的这些性质提供了一个自然的背景，它的Markov迹是一个关联图的色多项式。我们利用它证明了色多项式在Q={\phi}+1处的Tutte的另一恒等式来自Temperley-Lieb代数中的Jones-Wenzl投影仪。对于j<n个正整数，我们将此恒等式推广到每个值q=2+2\cos(2\pij/(n+1))。当j=1时，这些Q是Beraha数，Tutte推测存在这样的恒等式。本文给出了色多项式关系序列的一个递推公式。
---
英文标题：
《Tutte chromatic identities from the Temperley-Lieb algebra》
---
作者：
Paul Fendley and Vyacheslav Krushkal
---
最新提交年份：
2008
---
分类信息：

一级分类：Mathematics        数学
二级分类：Combinatorics        组合学
分类描述：Discrete mathematics, graph theory, enumeration, combinatorial optimization, Ramsey theory, combinatorial game theory
离散数学，图论，计数，组合优化，拉姆齐理论，组合对策论
--
一级分类：Physics        物理学
二级分类：Statistical Mechanics        统计力学
分类描述：Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变，热力学，场论，非平衡现象，重整化群和标度，可积模型，湍流
--
一级分类：Mathematics        数学
二级分类：Geometric Topology        几何拓扑
分类描述：Manifolds, orbifolds, polyhedra, cell complexes, foliations, geometric structures
流形，轨道，多面体，细胞复合体，叶状，几何结构
--
一级分类：Mathematics        数学
二级分类：Quantum Algebra        量子代数
分类描述：Quantum groups, skein theories, operadic and diagrammatic algebra, quantum field theory
量子群，skein理论，运算代数和图解代数，量子场论
--

---
英文摘要：
  This paper introduces a conceptual framework, in the context of quantum topology and the algebras underlying it, for analyzing relations obeyed by the chromatic polynomial \chi(Q) of planar graphs. Using it we give new proofs and substantially extend a number of classical results concerning the combinatorics of the chromatic polynomial. In particular, we show that Tutte's golden identity is a consequence of level-rank duality for SO(N) topological quantum field theories and Birman-Murakami-Wenzl algebras. This identity is a remarkable feature of the chromatic polynomial relating \chi({\phi+2}) for any triangulation of the sphere to (\chi({\phi+1}))^2 for the same graph, where \phi denotes the golden ratio. The new viewpoint presented here explains that Tutte's identity is special to these values of the parameter Q. A natural context for analyzing such properties of the chromatic polynomial is provided by the {\em chromatic algebra}, whose Markov trace is the chromatic polynomial of an associated graph. We use it to show that another identity of Tutte's for the chromatic polynomial at Q={\phi}+1 arises from a Jones-Wenzl projector in the Temperley-Lieb algebra. We generalize this identity to each value Q= 2+2\cos(2\pi j/(n+1)) for j< n positive integers. When j=1, these Q are the Beraha numbers, where the existence of such identities was conjectured by Tutte. We present a recursive formula for this sequence of chromatic polynomial relations.
---
PDF链接：
https://arxiv.org/pdf/711.0016