Exceptional Dehn surgery on hyperbolic 3-manifolds

在本项目中，我们重点研究了双曲三维流形上的特殊Dehn手术中的环状手术，并得到了一些结果。众所周知，对3-球面中的双曲纽结进行环形手术对应于一个整数或半个整数。首先，我们从桥指数的角度确定了非积分环面手术最简单的双曲纽结。也就是说，我们证明了(-2，3，7)-椒盐卷曲结是唯一具有非积分环状手术的三桥结。此外，在椒盐卷饼结中，只有这个结做了这样的手术。其次，我们提出了一个猜想：双曲纽结的任何积分环向斜率都有界于该纽结的亏格乘以4。在第一年，我们对两类重要的纽结，一类纽结和交错纽结，肯定地解决了这个猜想。在第二年，我们试图证明全部猜想，但我们不能这样做。但我们已经差不多完成了。根据本质环面与所附实体环面的核心的最小交数，我们得到了期望的上界，除非该上界不等于4。此外，环形外科手术的开口同时创造了一个克莱恩奶瓶。我们得到了克莱因瓶手术所有结的最好可能上界。利用这个引理，我们确定了环结点和2-桥结点的交叉数

In this project, we focused on toroidal surgery among exceptional Dehn surgery on hyperbolic 3-manifolds, and obtained several results. It is well known that a toroidal surgery of a hyperbolic knot in the 3-sphere corresponds to either an integer or a half-integer. First, we determined the simplest hyperbolic knot with non-integral toroidal surgery from a view of bridge index. That is, we showed that the (-2,3,7)-pretzel knot is the only 3-bridge knot with non integral toroidal surgery. Moreover, this knot is the only one with such surgery among pretzel knots. Second, we proposed a conjecture that any integral toroidal slope for a hyperbolic knot is bounded by the genus of the knot multiplied by four. In the first year, we solved this conjecture affirmatively for two important classes of knots, genus one knots and alternating knots. In the second year, we tried to prove the full conjecture, but we could not do this. But we have almost done. According to the minimal intersection number between an essential torus and the core of the attached solid torus, we got the expected upper bound, unless that number is not equal to four. Also, a toroidal surgery open creates a Klein bottle simultaneously. We got the best possible upper bound for Klein bottle surgery on all knots by using genera of knots. By using of this argument, we determined the crosscap numbers of torus knots and 2-bridge knots