[Paper Review] Hard squares on cylinders revisited

Michał Adamaszek|arXiv (Cornell University)|Feb 8, 2012

Topological and Geometric Data Analysis16 references3 citations

TL;DR

This paper investigates the homotopy types and Euler characteristics of independence complexes on cylindrical square grids, using recursive reductions and combinatorial models. It establishes new homotopy equivalences for $P_m \times C_n$ with even and odd circumferences, proves rational generating functions for the Witten index, and introduces a necklace model that conjecturally simplifies the structure of these invariants.

ABSTRACT

We consider the independence complexes of square grids with cylindrical boundary conditions. When one of the dimensions is small we use simple reductions induced by edge removals to show explicit natural homotopy equivalences between those spaces. In the second part we expand the results of Jonsson, who calculated the Euler characteristic of cylinders with odd circumference. We describe a series of results for cylinders of even circumference. Finally we define a completely independent combinatorial model (necklaces) which calculates the generating functions of the Euler characteristic of cylindrical grids. We conjecture that this model has some particularly simple structure.

Motivation & Objective

• To extend Jonsson's work on the Euler characteristic of hard squares on odd-circumference cylinders to even circumferences.
• To establish explicit homotopy equivalences for independence complexes of cylindrical grids $P_m \times C_n$ using edge-removal reductions.
• To develop a combinatorial necklace model that computes the generating functions of the Witten index for cylindrical grids.
• To conjecture a simple underlying structure in the necklace model that explains the rational generating functions of the Euler characteristic.

Proposed method

• Uses edge-removal reductions to derive recursive homotopy equivalences for $\mathrm{Ind}(P_m \times C_n)$, especially for small $m$ and even $n$.
• Applies suspension operations ($\Sigma$) to relate complexes of different sizes, yielding $\mathrm{Ind}(P_m \times C_n) \simeq \Sigma^k \mathrm{Ind}(P_m \times C_{n-d})$.
• Defines the Witten index $Z(G) = 1 - \chi(\mathrm{Ind}(G))$ to analyze the Euler characteristic recursively.
• Introduces a combinatorial necklace model to compute generating functions $f_n(t) = \sum_{m=0}^\infty Z(P_m \times C_n)t^m$ for even $n$.
• Analyzes the denominator structure of $f_n(t)$, showing all poles are roots of unity, and computes explicit rational forms for $n \leq 22$.
• Decomposes a directed graph $\mathrm{Neck}(k,n)$ into cycles to model the necklace structure and predict generating function behavior.

Experimental results

Research questions

• RQ1 What is the homotopy type of the independence complex $\mathrm{Ind}(P_m \times C_n)$ for cylindrical grids with even circumference $n$?
• RQ2 Can the Euler characteristic of $\mathrm{Ind}(P_m \times C_n)$ for even $n$ be computed via a recursive or generating function method?
• RQ3 Does the proposed necklace model provide a combinatorial explanation for the rational structure of the generating functions $f_n(t)$?
• RQ4 Is there a simple underlying structure in the necklace model that explains the observed periodicity and rationality of the Witten index?
• RQ5 How do the homotopy equivalences for even $n$ relate to those for odd $n$, especially under Thapper's conjecture?

Key findings

• New homotopy equivalences are established: $\mathrm{Ind}(P_2 \times C_n) \simeq \Sigma^2 \mathrm{Ind}(P_2 \times C_{n-4})$ and $\mathrm{Ind}(P_3 \times C_n) \simeq \Sigma^6 \mathrm{Ind}(P_3 \times C_{n-8})$ for even $n$, extending known results.
• The generating function $f_n(t)$ for $Z(P_m \times C_n)$ is rational with poles only at roots of unity, for all even $n$.
• Explicit rational forms are computed for $f_n(t)$ up to $n=22$, with denominators factored into cyclotomic polynomials $\Phi_d(t)$.
• The necklace model decomposes into cycles, and the cycle decomposition table (Table 3) reveals a structured pattern in the graph $\mathrm{Neck}(k,n)$ for $n \leq 36$, suggesting deeper algebraic structure.
• The numerator polynomials $q_n(t)$ for $f_n(t)$ are explicitly given for $n \leq 22$, with degrees ranging from 12 to 62 and negative leading coefficients.
• The paper conjectures that the necklace model has a particularly simple structure that explains the rationality and periodicity of the Witten index, though this remains unproven.