Hilbert polynomials of configuration spaces over graphs of circumference at most 1
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916768681820160 |
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| author | An, Byung Hee Kim, Jang Soo |
| author_facet | An, Byung Hee Kim, Jang Soo |
| contents | The $ k $-configuration space $ B_kΓ$ of a topological space $ Γ$ is the space of sets of $ k $ distinct points in $ Γ$. In this paper, we consider the case where $ Γ$ is a graph of circumference at most $1$. We show that for all $ k\ge0 $, the $ i $-th Betti number of $ B_kΓ$ is given by a polynomial $P_Γ^i(k)$ in $ k $, called the Hilbert polynomial of $ Γ$. We find an expression for the Hilbert polynomial $P_Γ^i(k)$ in terms of those coming from the canonical $1$-bridge decomposition of $ Γ$. We also give a combinatorial description of the coefficients of $P_Γ^i(k)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_24416 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hilbert polynomials of configuration spaces over graphs of circumference at most 1 An, Byung Hee Kim, Jang Soo Geometric Topology Algebraic Topology Combinatorics Primary: 20F36, 55R80, Secondary: 05C10, 13E15 The $ k $-configuration space $ B_kΓ$ of a topological space $ Γ$ is the space of sets of $ k $ distinct points in $ Γ$. In this paper, we consider the case where $ Γ$ is a graph of circumference at most $1$. We show that for all $ k\ge0 $, the $ i $-th Betti number of $ B_kΓ$ is given by a polynomial $P_Γ^i(k)$ in $ k $, called the Hilbert polynomial of $ Γ$. We find an expression for the Hilbert polynomial $P_Γ^i(k)$ in terms of those coming from the canonical $1$-bridge decomposition of $ Γ$. We also give a combinatorial description of the coefficients of $P_Γ^i(k)$. |
| title | Hilbert polynomials of configuration spaces over graphs of circumference at most 1 |
| topic | Geometric Topology Algebraic Topology Combinatorics Primary: 20F36, 55R80, Secondary: 05C10, 13E15 |
| url | https://arxiv.org/abs/2505.24416 |