Computer Assisted Projective Rigidity
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929460974977024 |
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| author | Daly, Charles |
| author_facet | Daly, Charles |
| contents | In this paper we provide a computer assisted proof that about two thousand surgeries far away from the ideal point in the hyperbolic Dehn filling space of the figure-eight knot complement are infinitesimally projectively rigid. We also prove that for projective deformations of the figure-eight knot complement sufficiently close to the complete hyperbolic structure, the induced map on the first cohomology of the longitude of the boundary torus is non-zero. This paper provides a complementary piece to the results of Heusener and Porti who showed that for each k in Z, there is a sufficiently large Nk for which every k/n-Dehn filling on the figure-eight knot complement for n larger than Nk is infinitesimally projectively rigid. In the process of the proof, we provide explicit representations of the figure-eight knot complement in PSO(3,1) which are rational in the real and imaginary parts of the shapes of the ideal tetrahedra used to glue the knot complement together. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_08405 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Computer Assisted Projective Rigidity Daly, Charles Geometric Topology Differential Geometry In this paper we provide a computer assisted proof that about two thousand surgeries far away from the ideal point in the hyperbolic Dehn filling space of the figure-eight knot complement are infinitesimally projectively rigid. We also prove that for projective deformations of the figure-eight knot complement sufficiently close to the complete hyperbolic structure, the induced map on the first cohomology of the longitude of the boundary torus is non-zero. This paper provides a complementary piece to the results of Heusener and Porti who showed that for each k in Z, there is a sufficiently large Nk for which every k/n-Dehn filling on the figure-eight knot complement for n larger than Nk is infinitesimally projectively rigid. In the process of the proof, we provide explicit representations of the figure-eight knot complement in PSO(3,1) which are rational in the real and imaginary parts of the shapes of the ideal tetrahedra used to glue the knot complement together. |
| title | Computer Assisted Projective Rigidity |
| topic | Geometric Topology Differential Geometry |
| url | https://arxiv.org/abs/2408.08405 |