由正方形和六角网格图构造的图的独立复形
Independence Complexes of Cylinders Constructed from Square and Hexagonal Grid Graphs
摘要：
Fendley，Schoutens和van Eerten [Fendley等人，J. Phys. A：Math. Gen.，38（2005），pp. 315-322]研究了负活性下的硬正方形模型。他们找到了分析和数值证据，证明具有周期性边界的传递矩阵的特征值都是单位根。他们还证明，对于一个m × n的正方形网格，具有双周期边界，当m和n是互素时，分拆函数等于1。这些结构后来由Jonsson [Jonsson，Electronic J. Combin.，13（1）（2006），R67]。在那里，人们还注意到，圆柱形的情况下，似乎有有趣的性质时，圆柱体的周长是奇数。特别地，当3是圆柱体的周长和宽度的除数减1时，分拆函数是-2。否则，它等于1。本文利用拓扑和组合的方法研究了活动为-1的硬正方形和硬六边形模型，具有单周期边界，即圆柱形识别。我们计算相关的独立复杂的小尺寸的同调群，并建议匹配，我们相信，与进一步的分析，可以帮助解决这个猜想。
英文摘要：
Fendley, Schoutens and van Eerten [Fendley et al., J. Phys. A: Math. Gen., 38 (2005), pp. 315-322] studied the hard square model at negative activity. They found analytical and numerical evidence that the eigenvalues of the transfer matrix with periodic boundary were all roots of unity. They also conjectured that for an m times n square grid, with doubly periodic boundary, the partition function is equal to 1 when m and n are relatively prime. These conjectures were later proven by Jonsson [Jonsson, Electronic J. Combin., 13(1) (2006), R67]. There, it was also noted that the cylindrical case seemed to have interesting properties when the circumference of the cylinder is odd. In particular, when 3 is a divisor of both the circumference and the width of the cylinder minus 1, the partition function is -2. Otherwise, it is equal to 1. In this paper, we investigate the hard square and hard hexagon models at activity -1, with single periodic boundary, i.e, cylindrical identifications, using both topological and combinatorial techniques. We compute the homology groups of the associated independence complex for small sizes and suggest a matching which, we believe, with further analysis could help solve the conjecture.
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