不能参数化地成为Legendrian的纽结族
Families of knots that cannot be made Legendrian parametrically
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中文总结 AI 辅助
本文证明对于每个n≥3，任何纽结类型的Legendrian代表及其形式代表在π_n层面上的群同态均非满射，并指出π_2层面的满射性依赖纽结类型，从而揭示了参数族在更高同伦层面的刚性。
中文摘要 AI 辅助
每个光滑纽结类型都承认一个Legendrian代表，这是接触拓扑中的一个经典结果。然而，类似的满射性问题在参数层面是开放的。在这项工作中，我们处理$n>1$的情况。我们证明对于每个$n\geq 3$，每个纽结类型$\mathcal K$，每个Legendrian代表$\mathcal L$和每个形式Legendrian代表$\mathcal{FL}$，相关的群同态$\pi_n(\mathcal{L})\to\pi_n(\mathcal{K})$和$\pi_n(\mathcal{FL})\to\pi_n(\mathcal{K})$永远不是满射的。然后我们证明在$\pi_2$层面的满射性取决于纽结类型。因此，这项工作证明了在$\pi_1$之外的每个更高同伦层面上，参数族存在刚性。
英文摘要
The fact that every smooth knot type admits a Legendrian representative is a classical result in contact topology. However, the analogous surjectivity question was open at the parametric level. In this work we address the $n>1$ case. We prove that for every $n\geq 3$, every knot type $\mathcal K$, every Legendrian representative $\mathcal L$ and every formal Legendrian representative $\mathcal{FL}$, the associated group homomorphisms $π_n(\mathcal{L})\toπ_n(\mathcal{K})$ and $π_n(\mathcal{FL})\toπ_n(\mathcal{K})$ are never surjective. We then show that surjectivity at the $π_2$-level depends on the knot type. This work thus proves the presence of rigidity for parametric families at every higher homotopy level beyond $π_1$.
