外科联系电话

Pub Date : 2024-06-06 DOI:10.4310/jsg.2023.v21.n6.a4
John Etnyre, Marc Kegel, Sinem Onaran
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引用次数: 0

摘要

众所周知,任何接触 3 美元-manifold 都可以通过在标准紧密接触 3 美元球中沿着 Legendrian 链接 $L$ 进行有理接触 Dehn 手术而获得。我们定义并研究了不同版本的接触手术数,即描述给定接触 3$-manifold的手术链接 $L$ 的最小分量数。众所周知,任何接触 3$-manifold都可以通过在标准紧密接触 3$-球中沿着 Legendrian 链接 $L$ 进行有理接触 Dehn 手术而得到。我们定义并研究了不同版本的接触手术数,即描述给定接触 3$-manifold的手术链接 $L$ 的最小分量数。在论文的第一部分,我们通过各种不等式将接触手术数与其他不变式联系起来。特别是,我们证明了接触流形的接触手术数从上而下受底层拓扑流形的拓扑手术数加三的约束。在第二部分,我们计算了 3 美元球面上所有接触结构的接触手术数。我们的结论是,在上述每个流形上都存在无限多的非异构接触结构,这些结构无法通过从标准紧密接触 3$球上的单一有理接触手术获得。我们进一步得到了 3$-torus和透镜空间的结果。作为上述结果证明的一个组成部分,我们将接触结构同向不变式的计算推广到具有更一般手术系数的接触手术上,这可能会引起独立的兴趣。
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Contact surgery numbers
It is known that any contact $3$-manifold can be obtained by rational contact Dehn surgery along a Legendrian link $L$ in the standard tight contact $3$-sphere. We define and study various versions of contact surgery numbers, the minimal number of components of a surgery link $L$ describing a given contact $3$-manifold under consideration. It is known that any contact $3$-manifold can be obtained by rational contact Dehn surgery along a Legendrian link $L$ in the standard tight contact $3$-sphere. We define and study various versions of contact surgery numbers, the minimal number of components of a surgery link $L$ describing a given contact $3$-manifold under consideration. In the first part of the paper, we relate contact surgery numbers to other invariants in terms of various inequalities. In particular, we show that the contact surgery number of a contact manifold is bounded from above by the topological surgery number of the underlying topological manifold plus three. In the second part, we compute contact surgery numbers of all contact structures on the $3$-sphere. Moreover, we completely classify the contact structures with contact surgery number one on $S^1 \times S^2$, the Poincaré homology sphere and the Brieskorn sphere $\Sigma(2,3,7)$.We conclude that there exist infinitely many non-isotopic contact structures on each of the above manifolds which cannot be obtained by a single rational contact surgery from the standard tight contact $3$-sphere. We further obtain results for the $3$-torus and lens spaces. As one ingredient of the proofs of the above results we generalize computations of the homotopical invariants of contact structures to contact surgeries with more general surgery coefficients which might be of independent interest.
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