On the Alexander polynomials of knots with Gordian distanceone
On the Alexander polynomials of knots with Gordian distanceone
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具有 Gordian 距离的纽结的亚历山大多项式
DOI：
10.1016/j.topol.2011.11.017
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发表时间：
2012
影响因子：
0.6
通讯作者：
Akio Kawauchi
中科院分区：
文献类型：
--
作者：
河合優年;他;Akio Kawauchi
本文考虑一对结点的亚历山大多项式可由一对距离为1的结点实现的条件。证明了在节点的亚历山大多项式的集合中存在无穷多个互不相交的无穷子集，使得每个子集中的每一对不同的元素都不能被任何一对距离为1的节点实现.作为子集之一，我们有一个包含三叶结和八字结的亚历山大多项式的无限集合。我们还表明，每对不同的亚历山大多项式，使之一是亚历山大多项式的切片节是可实现的一对结的戈尔迪距离1，使每对不同的元素组成的无限子集的亚历山大多项式的切片节是可实现的一对结的戈尔迪距离1。这些结果解决了Y. Nakanishi和我。阿钟
We consider a condition on a pair of the Alexander polynomials of knots which are realizable by a pair of knots with Gordian distance one. We show that there are infinitely many mutually disjoint infinite subsets in the set of the Alexander polynomials of knots such that every pair of distinct elements in each subset is not realizable by any pair of knots with Gordian distance one. As one of the subsets, we have an infinite set containing the Alexander polynomials of the trefoil knot and the figure eight knot. We also show that every pair of distinct Alexander polynomials such that one is the Alexander polynomial of a slice knot is realizable by a pair of knots of Gordian distance one, so that every pair of distinct elements in the infinite subset consisting of the Alexander polynomials of slice knots is realizable by a pair of knots with Gordian distance one. These results solve problems given by Y. Nakanishi and by I. Jong.