Dehn surgeries on knots which yield lens spaces and genera of knots
Dehn surgeries on knots which yield lens spaces and genera of knots
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Dehn 对结进行手术，产生晶状体空间和结属
DOI：
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发表时间：
1999
影响因子：
0.8
通讯作者：
M. Teragaito
中科院分区：
文献类型：
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作者：
H. Goda;M. Teragaito
这是一个有趣的开放性问题，当Dehn手术在3球S3中的结上可以产生透镜空间时（见[10，12]）。已经对特殊的纽结进行了一些研究;特别是，这个问题完全解决了环面纽结[21]和卫星纽结[3，29，31]。众所周知，有许多双曲结的例子允许Dehn手术产生透镜空间。例如，Fintushel和Stern [8]已经证明，在（-2，3，7）-椒盐卷饼结上的18-和19-手术分别给出透镜空间L（18，5）和L（19，7）。然而，双曲结的研究似乎没有取得实质性的进展。这可能是一些著名的双曲结（如2桥结[26]、交替结[5]）不允许手术产生透镜空间的原因。在本文中，我们专注于纽结的属有条不紊地对待目前的条件，并表明，有一个约束的基本组的顺序所产生的透镜空间Dehn手术上的双曲结。此外，这个新的观点使我们能够提出一个猜想，这样的约束，这对所有已知的例子。
It is an interesting open question when Dehn surgery on a knot in the 3-sphere S3 can produce a lens space (see [10, 12]). Some studies have been made for special knots; in particular, the question is completely solved for torus knots [21] and satellite knots [3, 29, 31]. It is known that there are many examples of hyperbolic knots which admit Dehn surgeries yielding lens spaces. For example, Fintushel and Stern [8] have shown that 18- and 19-surgeries on the (−2, 3, 7)-pretzel knot give lens spaces L(18, 5) and L(19, 7), respectively. However, there seems to be no essential progress on hyperbolic knots. It might be a reason that some famous classes of hyperbolic knots, such as 2-bridge knots [26], alternating knots [5], admit no surgery yielding lens spaces. In this paper we focus on the genera of knots to treat the present condition methodically and show that there is a constraint on the order of the fundamental group of the resulting lens space obtained by Dehn surgery on a hyperbolic knot. Also, this new standpoint enables us to present a conjecture concerning such a constraint, which holds for all known examples.