Classification of Alternating Knots with Tunnel Number One
Classification of Alternating Knots with Tunnel Number One
复制标题
一号隧道交错结的分类
DOI：
10.4310/cag.2005.v13.n1.a5
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发表时间：
2001
影响因子：
0.7
通讯作者：
M. Lackenby
中科院分区：
文献类型：
--
作者：
M. Lackenby
交替图编码了关于结的大量信息。例如，如果交替结是复合的，从图[10]中可以明显看出这一点。也可直接读出其属（[3],[12]）和杂交数（[7],[13],[17]）。在本文中，我们将这一原理应用于具有隧道1的交替结。回想一下，如果一个结点K有一个解结的隧道，那么它就有一个隧道1，这个隧道被定义为一个弧t，它被适当地嵌入到结点的外部使得S−int（N (K∪t)）是一个柄体。一般来说，确定一个给定的结是否有一号隧道，如果有，确定所有解结的隧道，是一个非常困难的问题。在本文中，我们给出了一个完整的分类与隧道1交替结，和所有他们的解开隧道，直到一个环境同位素的结外部。
An alternating diagram encodes a lot of information about a knot. For example, if an alternating knot is composite, this is evident from the diagram [10]. Also, its genus ([3], [12]) and its crossing number ([7], [13], [17]) can be read off directly. In this paper, we apply this principle to alternating knots with tunnel number one. Recall that a knot K has tunnel number one if it has an unknotting tunnel, which is defined to be an arc t properly embedded in the knot exterior such that S − int(N (K ∪ t)) is a handlebody. It is in general a very difficult problem to determine whether a given knot has tunnel number one, and if it has, to determine all its unknotting tunnels. In this paper, we give a complete classification of alternating knots with tunnel number one, and all their unknotting tunnels, up to an ambient isotopy of the knot exterior.