Khovanov同调与Potts模型的注记
Remarks on Khovanov Homology and the Potts Model
摘要：
本文献给奥列格·维罗60岁生日。本文讨论Khovanov同调及其与统计力学模型如Ising模型和Potts模型的关系。我们给出了一个相对独立的介绍Khovanov同调，也是Potts模型的一个括号状态和扩展的结图K（G）有关的一个可平面图G通过中间结构。我们认为原来的Khovanov同源性，也是由Stosic通过二色多项式定义的同源性，并检查这些值的Potts模型的配分拆函数可以表示在同调欧拉示性数。这些点出现在假想的温度下。本文的最后一部分通过Wick旋转将Potts模型在虚温度下重新表述为量子振幅，并使用相同的形式主义给出了一种新的量子算法，用于计算位于复平面单位圆上的变量值的Jones多项式。
英文摘要：
This paper is dedicated to Oleg Viro on his 60-th birthday. The paper is about Khovanov homology and its relationships with statistical mechanics models such as the Ising model and the Potts model. We give a relatively self-contained introduction to Khovanov homology, and also a reformulation of the Potts model in terms of a bracket state sum expansion on a knot diagram K(G) related to a planar graph G via the medial construction. We consider the original Khovanov homology and also the homology defined by Stosic via the dichromatic polynomial, and examine those values of the Potts model where the partition function can be expressed in terms of homological Euler characteristics. These points occur at imaginary temperature. The last part of the paper reformulates, via a Wick rotation, the Potts model at imaginary temperature as a quantum amplitude and uses the same formalism to give a new quantum algorithm for the Jones polynomial for values of the variable that lie on the unit circle in the complex plane.
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