论文标题
接触手术数字
Contact surgery numbers
论文作者
论文摘要
众所周知，可以通过标准紧密接触3-Sphere中的Legendrian Link L沿Legendrian Link L沿Legendrian Link l进行理性接触Dehn手术获得任何接触。我们定义和研究各种接触手术数字，即手术链接的最小数量，描述了所考虑的给定接触3个manifold。在本文的第一部分中，我们将接触手术数量与其他不平等现象相关联。特别是，我们表明，接触歧管的接触手术数量与下面的拓扑歧管的拓扑手术数量加上三个。在第二部分中，我们计算三个球体上所有接触结构的接触手术数量。此外，我们将接触结构的第一款与$ S^1 \ times S^2 $，Poincaré同源性领域和Brieskorn Sphere $σ（2,3,7）$分类。我们得出的结论是，上述每个流形上存在无限的许多非同位素接触结构，这些接触结构无法通过标准紧密的接触$ 3 $ -SPHERE获得的单个合理接触手术获得。我们进一步获得了3道齿和镜头空间的结果。作为上述结果证明的一种成分，我们将接触结构的同位不变性的计算概括为与可能具有独立感兴趣的更多一般手术系数接触手术。
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 $Σ(2,3,7)$. 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.
