On the Legendrian realisation of parametric families of knots
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914827827412992 |
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| author | Martínez-Aguinaga, Javier |
| author_facet | Martínez-Aguinaga, Javier |
| contents | We study the natural inclusion of the space of Legendrian embeddings in $(\mathbb{S}^3,ξ_{\operatorname{std}})$ into the space of smooth embeddings from a homotopical viewpoint.
T. Kálmán posed in [Kal] the open question of whether for every fixed knot type $\mathcal{K}$ and Legendrian representative $\mathcal{L}$, the homomorphism $π_1(\mathcal{L})\toπ_1(\mathcal{K})$ is surjective. We positively answer this question for infinitely many knot types $\mathcal{K}$ in the three main families (hyperbolic, torus and satellites) and every stabilised Legendrian representative in $(\mathbb{S}^3,ξ_{\operatorname{std}})$.
We then show that for every $n\geq 3$, the homomorphisms $π_n(\mathcal{L})\toπ_n(\mathcal{K})$ and $π_n(\mathcal{FL})\toπ_n(\mathcal{K})$ are never surjective for any knot type $\mathcal K$, Legendrian representative $\mathcal L$ or formal Legendrian representative $\mathcal{FL}$. This shows the existence of rigidity at every higher-homotopy level beyond $π_3$. For completeness, we also show that surjectivity at the $π_2$-level depends on the smooth knot type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_04293 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Legendrian realisation of parametric families of knots Martínez-Aguinaga, Javier Geometric Topology Differential Geometry Symplectic Geometry 53D10 We study the natural inclusion of the space of Legendrian embeddings in $(\mathbb{S}^3,ξ_{\operatorname{std}})$ into the space of smooth embeddings from a homotopical viewpoint. T. Kálmán posed in [Kal] the open question of whether for every fixed knot type $\mathcal{K}$ and Legendrian representative $\mathcal{L}$, the homomorphism $π_1(\mathcal{L})\toπ_1(\mathcal{K})$ is surjective. We positively answer this question for infinitely many knot types $\mathcal{K}$ in the three main families (hyperbolic, torus and satellites) and every stabilised Legendrian representative in $(\mathbb{S}^3,ξ_{\operatorname{std}})$. We then show that for every $n\geq 3$, the homomorphisms $π_n(\mathcal{L})\toπ_n(\mathcal{K})$ and $π_n(\mathcal{FL})\toπ_n(\mathcal{K})$ are never surjective for any knot type $\mathcal K$, Legendrian representative $\mathcal L$ or formal Legendrian representative $\mathcal{FL}$. This shows the existence of rigidity at every higher-homotopy level beyond $π_3$. For completeness, we also show that surjectivity at the $π_2$-level depends on the smooth knot type. |
| title | On the Legendrian realisation of parametric families of knots |
| topic | Geometric Topology Differential Geometry Symplectic Geometry 53D10 |
| url | https://arxiv.org/abs/2406.04293 |