{"paper":{"title":"Hilbert polynomials of configuration spaces over graphs of circumference at most 1","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.CO"],"primary_cat":"math.GT","authors_text":"Byung Hee An, Jang Soo Kim","submitted_at":"2025-05-30T09:54:45Z","abstract_excerpt":"The $ k $-configuration space $ B_k\\Gamma $ of a topological space $ \\Gamma $ is the space of sets of $ k $ distinct points in $ \\Gamma $. In this paper, we consider the case where $ \\Gamma $ is a graph of circumference at most $1$. We show that for all $ k\\ge0 $, the $ i $-th Betti number of $ B_k\\Gamma $ is given by a polynomial $P_\\Gamma^i(k)$ in $ k $, called the Hilbert polynomial of $ \\Gamma $. We find an expression for the Hilbert polynomial $P_\\Gamma^i(k)$ in terms of those coming from the canonical $1$-bridge decomposition of $ \\Gamma $. We also give a combinatorial description of the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.24416","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2505.24416/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}