The Local-Global Conjecture is False for Generalized Circle Packings
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arXiv
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| Format: | Preprint |
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2025
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| author | Shi, Hanqi Shi, Wenyuan Whitehead, Ian Williams-Tracy, Ham Zhang, Jeffrey Zhirui |
| author_facet | Shi, Hanqi Shi, Wenyuan Whitehead, Ian Williams-Tracy, Ham Zhang, Jeffrey Zhirui |
| contents | Haag, Kertzer, Rickards, and Stange disprove the Local-Global Conjecture for Apollonian circle packings. We extend their disproof to four more types of integral circle packing: the octahedral, cubic, square, and triangular packings. In each case, we find quadratic invariants which imply quadratic reciprocity obstructions to the conjecture in certain packings. We utilize an explicit parametrization of circles tangent to a fixed circle in each packing type, and a quadratic reciprocity argument. Even in the packings where we do not find quadratic obstructions, the curvatures exhibit a predictable reciprocity structure. This leads to partial obstructions on integers appearing as curvatures in subsets of the packing. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_21702 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Local-Global Conjecture is False for Generalized Circle Packings Shi, Hanqi Shi, Wenyuan Whitehead, Ian Williams-Tracy, Ham Zhang, Jeffrey Zhirui Number Theory Primary: 52C26, 11D99, Secondary: 20H10, 11E12, 11A15, 11B99 Haag, Kertzer, Rickards, and Stange disprove the Local-Global Conjecture for Apollonian circle packings. We extend their disproof to four more types of integral circle packing: the octahedral, cubic, square, and triangular packings. In each case, we find quadratic invariants which imply quadratic reciprocity obstructions to the conjecture in certain packings. We utilize an explicit parametrization of circles tangent to a fixed circle in each packing type, and a quadratic reciprocity argument. Even in the packings where we do not find quadratic obstructions, the curvatures exhibit a predictable reciprocity structure. This leads to partial obstructions on integers appearing as curvatures in subsets of the packing. |
| title | The Local-Global Conjecture is False for Generalized Circle Packings |
| topic | Number Theory Primary: 52C26, 11D99, Secondary: 20H10, 11E12, 11A15, 11B99 |
| url | https://arxiv.org/abs/2510.21702 |