Bridge indices of spatial graphs and diagram colorings
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917126097338368 |
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| author | Blackwell, Sarah Pongtanapaisan, Puttipong Vo, Hanh |
| author_facet | Blackwell, Sarah Pongtanapaisan, Puttipong Vo, Hanh |
| contents | We extend the Wirtinger number of links, an invariant originally defined by Blair, Kjuchukova, Velazquez, and Villanueva in terms of extending initial colorings of some strands of a diagram to the entire diagram, to spatial graphs. We prove that the Wirtinger number equals the bridge index of spatial graphs, and we implement an algorithm in Python which gives a more efficient way to estimate upper bounds of bridge indices. Combined with lower bounds from diagram colorings by elements from certain algebraic structures and clasping techniques, we obtain exact bridge indices for a large family of almost unknotted spatial graphs. We also show that for every possible negative Euler characteristic, there exist almost unknotted graphs of arbitrarily large bridge index. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_23253 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bridge indices of spatial graphs and diagram colorings Blackwell, Sarah Pongtanapaisan, Puttipong Vo, Hanh Geometric Topology Combinatorics 57M15 (Primary), 05C10 We extend the Wirtinger number of links, an invariant originally defined by Blair, Kjuchukova, Velazquez, and Villanueva in terms of extending initial colorings of some strands of a diagram to the entire diagram, to spatial graphs. We prove that the Wirtinger number equals the bridge index of spatial graphs, and we implement an algorithm in Python which gives a more efficient way to estimate upper bounds of bridge indices. Combined with lower bounds from diagram colorings by elements from certain algebraic structures and clasping techniques, we obtain exact bridge indices for a large family of almost unknotted spatial graphs. We also show that for every possible negative Euler characteristic, there exist almost unknotted graphs of arbitrarily large bridge index. |
| title | Bridge indices of spatial graphs and diagram colorings |
| topic | Geometric Topology Combinatorics 57M15 (Primary), 05C10 |
| url | https://arxiv.org/abs/2410.23253 |