Heegaard Floer homology and alternating knots.
Heegaard Floer homology and alternating knots.
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Heegaard Floer 同源性和交替结。
DOI：
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
Z. Szabó
中科院分区：
文献类型：
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作者：
P. Ozsváth;Z. Szabó
在前面的一篇文章中，我们引入了定向三维流形中零同调纽结K的一个纽结不变量，它与Y的Heegaard Floer同调密切相关。本文研究了三球面上纽结的这些纽结同调群的一些性质。我们给出了链复合体的生成元及其分次的组合描述。借助于这一描述，我们确定了交错纽结的纽结同调，表明在这种特殊情况下，它仅取决于纽结的签名和Alexander多项式(推广了Rasmussen关于两桥纽结的一个结果)。应用包括对交替纽结的Alexander多项式的新限制。
In an earlier paper, we introduced a knot invariant for a null-homologous knot K in an oriented three-manifold Y , which is closely related to the Heegaard Floer homology of Y . In this paper we investigate some properties of these knot homology groups for knots in the three-sphere. We give a combinatorial description for the generators of the chain complex and their gradings. With the help of this description, we determine the knot homology for alternating knots, showing that in this special case, it depends only on the signature and the Alexander polynomial of the knot (generalizing a result of Rasmussen for two-bridge knots). Applications include new restrictions on the Alexander polynomial of alternating knots.