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On the Alexander polynomials of knots with Gordian distance one

机译：关于结点为1的结的亚历山大多项式

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摘要

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.
机译：我们考虑一对结的亚历山大多项式的条件，这些条件可以通过一对距离为Gordian的结来实现。我们表明，结的亚历山大多项式集合中有无限多个相互不相交的无限子集，这样每个子集中的每对不同元素都无法通过高迪安距离为1的任何一对结来实现。作为子集之一，我们有一个无限集，其中包含三叶形结和八字形结的亚历山大多项式。我们还表明，每对不同的Alexander多项式（例如一个切片结的Alexander多项式）可以通过一对Gordian距离为1的结来实现，因此无限子集中的每对不同元素都由A的Alexander多项式组成切片结可通过一对距离为Gordian的结来实现。这些结果解决了Y. Nakanishi和I. Jong提出的问题。

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