Connecting essential triangulations I: via 2-3 and 0-2 moves
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arXiv
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| Format: | Preprint |
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2024
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| author | Kalelkar, Tejas Schleimer, Saul Segerman, Henry |
| author_facet | Kalelkar, Tejas Schleimer, Saul Segerman, Henry |
| contents | Suppose that $M$ is a compact, connected three-manifold with boundary. We show that if the universal cover has infinitely many boundary components then $M$ has an ideal triangulation which is essential: no edge can be homotoped into the boundary. Under the same hypotheses, we show that the set of essential triangulations of $M$ is connected via 2-3, 3-2, 0-2, and 2-0 moves.
The above results are special cases of our general theory. We introduce $L$-essential triangulations: boundary components of the universal cover receive labels and no edge has the same label at both ends. As an application, under mild conditions on a representation, we construct an ideal triangulation for which a solution to Thurston's gluing equations recovers the given representation.
Our results also imply that such triangulations are connected via 2-3, 3-2, 0-2, and 2-0 moves. Together with results of Pandey and Wong, this proves that Dimofte and Garoufalidis' 1-loop invariant is independent of the choice of essential triangulation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_03539 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Connecting essential triangulations I: via 2-3 and 0-2 moves Kalelkar, Tejas Schleimer, Saul Segerman, Henry Geometric Topology 57K16, 57K31, 57K32, 57Q15 Suppose that $M$ is a compact, connected three-manifold with boundary. We show that if the universal cover has infinitely many boundary components then $M$ has an ideal triangulation which is essential: no edge can be homotoped into the boundary. Under the same hypotheses, we show that the set of essential triangulations of $M$ is connected via 2-3, 3-2, 0-2, and 2-0 moves. The above results are special cases of our general theory. We introduce $L$-essential triangulations: boundary components of the universal cover receive labels and no edge has the same label at both ends. As an application, under mild conditions on a representation, we construct an ideal triangulation for which a solution to Thurston's gluing equations recovers the given representation. Our results also imply that such triangulations are connected via 2-3, 3-2, 0-2, and 2-0 moves. Together with results of Pandey and Wong, this proves that Dimofte and Garoufalidis' 1-loop invariant is independent of the choice of essential triangulation. |
| title | Connecting essential triangulations I: via 2-3 and 0-2 moves |
| topic | Geometric Topology 57K16, 57K31, 57K32, 57Q15 |
| url | https://arxiv.org/abs/2405.03539 |