Symmetry groups of flat fully augmented links and their complements
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908704824098816 |
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| author | Millichap, Christian Trapp, Rolland |
| author_facet | Millichap, Christian Trapp, Rolland |
| contents | In this paper, we prove that the (orientation-preserving) symmetry groups of $b$-prime flat fully augmented links correspond exactly with the finite subgroups of $O(3)$. We accomplish this by first developing a dictionary between automorphisms of a $3$-connected planar cubic graph associated to a flat fully augmented link $L$ and orientation-preserving symmetries of $L$. Our work also provides a simple method to explicitly construct infinite classes of distinct $b$-prime flat fully augmented links $\{L_i\}$ with $Sym^{+}(\mathbb{S}^{3} \setminus L_i) \cong Sym^{+}(\mathbb{S}^{3}, L_i) \cong G$, for any $G$ that is a finite subgroup of $O(3)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_10050 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Symmetry groups of flat fully augmented links and their complements Millichap, Christian Trapp, Rolland Geometric Topology 57K10, 57K32 In this paper, we prove that the (orientation-preserving) symmetry groups of $b$-prime flat fully augmented links correspond exactly with the finite subgroups of $O(3)$. We accomplish this by first developing a dictionary between automorphisms of a $3$-connected planar cubic graph associated to a flat fully augmented link $L$ and orientation-preserving symmetries of $L$. Our work also provides a simple method to explicitly construct infinite classes of distinct $b$-prime flat fully augmented links $\{L_i\}$ with $Sym^{+}(\mathbb{S}^{3} \setminus L_i) \cong Sym^{+}(\mathbb{S}^{3}, L_i) \cong G$, for any $G$ that is a finite subgroup of $O(3)$. |
| title | Symmetry groups of flat fully augmented links and their complements |
| topic | Geometric Topology 57K10, 57K32 |
| url | https://arxiv.org/abs/2512.10050 |