Independence Complexes of Cylinders Constructed from Square and Hexagonal Grid Graphs

In the paper [Fendley et al., J. Phys. A: Math. Gen., 38 (2005), pp. 315-322], Fendley, Schoutens and van Eerten 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 × n square grid, with doubly periodic boundary, the partition function is equal to 1 when m and n are relatively prime. These conjectures were proven in [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. We also briefly review a technique recently described by Bousquet-Mélou, Linusson and Nevo, for determining some of the eigenvalues of the transfer matrix of the hard square model with cylindrical identification using a related, but more easily analysed model.