Non-hyperbolic Dehn surgeries on hyperbolic knots

瑟斯顿的双曲Dehn手术定理断言：如果3-球面上的一个纽结K是双曲的(即，K的补集允许有一个有限体积的完全双曲结构)，则K上的所有Dehn手术都产生双曲3-流形。然后，描述双曲纽结上的非双曲手术是很重要的。众所周知，任何非双曲手术都是简化手术、环状手术、Seifert纤维手术或产生反例几何猜想的手术。在本研究中，我们集中于Seifert纤维手术。Seifert纤维环结手术可以通过考虑Seifert外部纤维如何延伸到手术流形上来自然地解释。Seifert光纤手术在双曲线结上有什么自然的解释吗？Berge给出了一个显式结构，它产生了几个无限族纽结，每个纽结允许一个透镜空间Dehn手术.Dean引入了一个原始/Seifert-fibered结构，它是Berge结构的自然修改，并提供了每个纽结允许Seifert-fibered手术的无限族.我们确定了非双曲的原始/Seifert纤维结，并证明了对于每个这样的结，任何积分的小Seifert运算都产生于原始/Seifert纤维结构。我们还研究了双曲纽结上的纵向例外手术。Gabai在3-球面上发现了一个双曲纤维纽结，而纵向手术在其上产生了环状流形，现在已知存在无限多个这样的双曲纤维纽结。另一方面，还没有已知的双曲线纤维结与纵向的塞弗特纤维手术的例子，Teragaito问是否没有这样的例子。我们通过构造一个双曲纤维结的无限族给出了这个问题的答案，每个双曲纤维结允许一个纵向的塞弗特纤维手术。

Thurston's hyperbolic Dehn surgery theorem asserts that if a knot K in the 3-sphere is hyperbolic (i.e., the complement of K admits a complete hyperbolic structure of finite volume), then all but finitely many Dehn surgeries on K yield hyperbolic 3-manifolds. Then it is important to describe non-hyperbolic surgeries on hyperbolic knots. It is known that any non-hyperbolic surgery is a reducing surgery, a toroidal surgery, a Seifert fibered surgery, or a surgery producing a counterexample to Geometrization conjecture.In this research we focus on Seifert fibered surgeries. Seifert fibered surgeries on torus knots can be naturally explained by considering how Seifert fibrations of the exterior extends over the surgered manifolds. Are there any natural explanation for Seifert fibered surgeries on hyperbolic knots? Berge gave an explicit construction which yields several infinite families of knots each admitting a lens space Dehn surgery.Dean introduced a primitive/Seifert-fibered construction, which is a natural modification of Berge's construction and provides infinite families of knots each of which admits Seifert fibered surgery. We determine non-hyperbolic, primitive/Seifert-fibered knots and show that for each such knots any integral, small Seifert surgery arises from a primitive/Seifert-fibered construction. We also study longitudinal exceptional surgeries on hyperbolic knots.Gabai found a hyperbolic, fibered knot in the 3-sphere on which a longitudinal surgery produces a toroidal manifold, and now it is known that there are infinitely many such hyperbolic, fibered knots. On the other hand, there have been no known examples of hyperbolic, fibered knots with longitudinal, Seifert fibered surgeries, and Teragaito asks if there are no such examples. We give an answer this question by constructing an infinite family of hyperbolic, fibered knots each of which admits a longitudinal, Seifert fibered surgery.