English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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. Theyfound analytical and numerical evidence that the eigenvalues of the transfermatrix with periodic boundary were all roots of unity. They also conjecturedthat for an m times n square grid, with doubly periodic boundary, the partitionfunction is equal to 1 when m and n are relatively prime. These conjectureswere later proven by Jonsson [Jonsson, Electronic J. Combin., 13(1) (2006),R67]. There, it was also noted that the cylindrical case seemed to haveinteresting properties when the circumference of the cylinder is odd. Inparticular, when 3 is a divisor of both the circumference and the width of thecylinder minus 1, the partition function is -2. Otherwise, it is equal to 1. Inthis paper, we investigate the hard square and hard hexagon models at activity-1, with single periodic boundary, i.e, cylindrical identifications, using bothtopological and combinatorial techniques. We compute the homology groups of theassociated independence complex for small sizes and suggest a matching which,we believe, with further analysis could help solve the conjecture.

Independence Complexes of Cylinders Constructed from Square and Hexagonal Grid Graphs