On Floer homology and the Berge conjecture on knots admitting lens space surgeries
On Floer homology and the Berge conjecture on knots admitting lens space surgeries
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DOI：
10.1090/s0002-9947-2010-05117-7
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发表时间：
2007-10
影响因子：
1.3
通讯作者：
M. Hedden
中科院分区：
文献类型：
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作者：
M. Hedden
我们完成了由贝克，格雷斯比，和作者提出的两部分计划的第一步，以证明Berge的建设的结在三个领域承认透镜空间手术是完整的。第一步，我们在这里证明，是要表明，一个结在一个透镜空间与threesperson手术有简单的（在意义上的秩）结弗洛尔同源。第二步（简化）涉及到显示，对于一个固定的透镜空间，只有简单的弗洛尔同调结属于一个简单的有限家庭。使用贝克的结果，我们提供了证据的程序的结构部分，表明它适用于某个家庭的结。再加上工作的Ni，这些结提供了第一个无限家庭的非平凡的结，其特点是他们的结弗洛尔同源。作为另一个应用，我们提供了Berge定理的Floer同调证明。
We complete the first step in a two-part program proposed by Baker, Grigsby, and the author to prove that Berge’s construction of knots in the three-sphere which admit lens space surgeries is complete. The first step, which we prove here, is to show that a knot in a lens space with a threesphere surgery has simple (in the sense of rank) knot Floer homology. The second (conjectured) step involves showing that, for a fixed lens space, the only knots with simple Floer homology belong to a simple finite family. Using results of Baker, we provide evidence for the conjectural part of the program by showing that it holds for a certain family of knots. Coupled with work of Ni, these knots provide the first infinite family of non-trivial knots which are characterized by their knot Floer homology. As another application, we provide a Floer homology proof of a theorem of Berge.