# Configuration spaces of hard squares in a rectangle

- Published in 2020
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We study the configuration spaces C(n;p,q) of n labeled hard squares in a p by q rectangle, a generalization of the well-known "15 Puzzle". Our main interest is in the topology of these spaces. Our first result is to describe a cubical cell complex and prove that is homotopy equivalent to the configuration space. We then focus on determining for which n, j, p, and q the homology group $H_j [ C(n;p,q) ]$ is nontrivial. We prove three homology-vanishing theorems, based on discrete Morse theory on the cell complex. Then we describe several explicit families of nontrivial cycles, and a method for interpolating between parameters to fill in most of the picture for "large-scale" nontrivial homology.

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### BibTeX entry

@article{Configurationspacesofhardsquaresinarectangle, title = {Configuration spaces of hard squares in a rectangle}, abstract = {We study the configuration spaces C(n;p,q) of n labeled hard squares in a p by q rectangle, a generalization of the well-known "15 Puzzle". Our main interest is in the topology of these spaces. Our first result is to describe a cubical cell complex and prove that is homotopy equivalent to the configuration space. We then focus on determining for which n, j, p, and q the homology group {\$}H{\_}j [ C(n;p,q) ]{\$} is nontrivial. We prove three homology-vanishing theorems, based on discrete Morse theory on the cell complex. Then we describe several explicit families of nontrivial cycles, and a method for interpolating between parameters to fill in most of the picture for "large-scale" nontrivial homology.}, url = {http://arxiv.org/abs/2010.14480v1 http://arxiv.org/pdf/2010.14480v1}, year = 2020, author = {Hannah Alpert and Ulrich Bauer and Matthew Kahle and Robert MacPherson and Kelly Spendlove}, comment = {}, urldate = {2020-11-06}, archivePrefix = {arXiv}, eprint = {2010.14480}, primaryClass = {math.AT}, collections = {puzzles} }