Non-hyperbolic Dehn surgeries on hyperbolic knots

Thurston的双曲Dehn手术定理断言，如果3-球面中的纽结K是双曲的（即，K的补允许有限体积的完全双曲结构），则K上几乎所有的Dehn运算都产生双曲3-流形。然后，重要的是描述双曲结上的非双曲手术。众所周知，任何非双曲线手术都是缩小手术、环形手术、塞弗特纤维手术或产生几何化猜想反例的手术。塞弗特纤维手术环面结可以很自然地解释考虑塞弗特纤维的外部延伸surgered流形。有没有任何自然的解释塞弗特纤维手术的双曲结？Berge给出了一个明确的建设，产生几个无限的家庭结每个承认一个透镜空间Dehn手术。据证实，任何透镜空间手术都可以 ...更多信息 Berge的原始/原始结构解释。原始/原始构造的自然推广是由迪恩完成的，这解释了许多塞弗特在双曲结上的纤维手术。在1996年，戈登提出任何塞弗特纤维手术都可以用迪恩的原始/塞弗特纤维结构来解释。最近Eudave-Munoz已经表明，所有已知的由Montesinos技巧构建的塞弗特纤维外科手术的例子都可以用原始/塞弗特纤维结构来解释。在这项研究中，作为与托马斯马特曼和Katura宫崎的联合工作，我们构建了两个无限的结族，每个结族都允许塞弗特纤维手术，而这些手术都不是来自迪恩的原始/塞弗特纤维结构。这否定了一个猜想，即所有塞弗特纤维外科手术都源于迪恩的原始/塞弗特纤维结构。（-3，3，5）-椒盐卷饼结属于这两个无限族。最近，我们感兴趣的问题：如果K有一个塞弗特纤维手术，那么K嵌入在亏格2 Heegaard表面的3球？少

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 counter example 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 manifold. 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. It is conjectured that any lens space surgery can be … More explained by Berge's primitive/primitive construction. The natural generalization of primitive /primitive construction was done by Dean, which explains many Seifert fibered surgeries on hyperbolic knots. In 1996, Gordon conjectured that any Seifert fibered surgery can be explained by Dean's primitive/Seifert-fibered construction. Recently Eudave-Munoz has shown that all known examples of Seifert fibered surgeries constructed by the Montesinos trick can be explained by primitive/Seifert-fibered construction. In this research, as a joint work with Thomas Mattman and Katura Miyazaki, we construct two infinite families of knots each of which admits a Seifert fibered surgery with none of these surgeries coming from Dean' s primitive/Seifert-fibered construction. This disproves a conjecture that all Seifert fibered surgeries arise from Dean' s primitive/Seifert-fibered construction. The (-3, 3, 5)-pretzel knot belongs to both of the infinite families. Very recently we are interested in the questions : If K has a Seifert fibered surgery, then is K embedded in a genus 2 Heegaard surface of the 3-sphere? Less