The topological complexity of the ordered configuration space of disks in a strip
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912926731862016 |
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| author | Wawrykow, Nicholas |
| author_facet | Wawrykow, Nicholas |
| contents | How hard is it to program $n$ robots to move about a long narrow aisle such that only $w$ of them can fit across the width of the aisle? In this paper, we answer that question by calculating the topological complexity of $\text{conf}(n,w)$, the ordered configuration space of open unit-diameter disks in the infinite strip of width $w$. By studying its cohomology ring, we prove that, as long as $n$ is greater than $w$, the topological complexity of $\text{conf}(n,w)$ is $2n-2\big\lceil\frac{n}{w}\big\rceil+1$, providing a lower bound for the minimum number of cases such a program must consider. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_11711 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The topological complexity of the ordered configuration space of disks in a strip Wawrykow, Nicholas Algebraic Topology Differential Geometry 55R80, 57Z20 How hard is it to program $n$ robots to move about a long narrow aisle such that only $w$ of them can fit across the width of the aisle? In this paper, we answer that question by calculating the topological complexity of $\text{conf}(n,w)$, the ordered configuration space of open unit-diameter disks in the infinite strip of width $w$. By studying its cohomology ring, we prove that, as long as $n$ is greater than $w$, the topological complexity of $\text{conf}(n,w)$ is $2n-2\big\lceil\frac{n}{w}\big\rceil+1$, providing a lower bound for the minimum number of cases such a program must consider. |
| title | The topological complexity of the ordered configuration space of disks in a strip |
| topic | Algebraic Topology Differential Geometry 55R80, 57Z20 |
| url | https://arxiv.org/abs/2404.11711 |