On the divisibility of knot groups.
On the divisibility of knot groups.
复制标题
关于结群的可分性。
DOI：
10.2140/pjm.1974.52.491
复制
发表时间：
1974
影响因子：
0.6
通讯作者：
K. Murasugi
中科院分区：
文献类型：
--
作者：
K. Murasugi
一个群G称为R-群，如果对于每对元素x和y，以及每一个自然数nf，从xn = y可以得出x = y。换句话说，G是群，如果G对每个方程x = a都只有一个解。如果G是i-群，则G是局部无限的。然而，即使G仅限于S中的一个纽结的群，反之也不必成立。例如，设G是K（m，ri）的群，即（m，n）型环面纽结。G有表示G =（α，B：a = B）。则方程x = a有无穷多个不同的解，x = a，（ba）a（ba）"| f（ba）a（ba）~，这一观察立即给出了[3]中问题N的否定答案. Neutroth询问是否可以订购结组。实际上，K {m，n}（\m|,| n| ^2）不能被排序，因为一个有序群总是一个i？组因此，问题N现在引出了一个稍微弱一点的问题：除了环面纽结群之外的纽结群可以被排序吗？或者，除了环面结群之外的结群是iZ群吗？本文给出了一个单纽结的群是单纽结群的一个充分条件。(See定理2）利用这个条件，我们可以证明，例如，8字形纽结的群是一个i2-群。(See第3项或第5项提案）。
A group G is called an R-group if for every pair of elements x and y, and every natural number nf it follows from x n = y that x = y. In other words, G is an ϋϊ-group if G has not more than one solution for every equation x = a. If G is an iϋ-group, G is locally infinite. The converse, however, need not be true even if G is restricted to the group of a knot in S. For example, let G be the group of K(m, ri), the torus knot of type (m, n). G has a presentation G = (α, b: a = b). Then the equation x = a has infinitely many distinct solutions, x = a, (ba)a(ba)"ιf (ba) a(ba)~, This observation gives immediately a negative answer to Problem N in [3]. Neuwirth asks if a knot group can be ordered. In fact, the group of K{m, n)(\ m |, | n | ^ 2) cannot be ordered, since an ordered group is always an i?-group. Therefore, Problem N now leads slightly weaker problems: Can a knot group other than torus knot groups be ordered? Or, is a knot group other than torus knot groups an iZ-group? The purpose of this paper is to give a sufficient condition for the group of a fibred knot to be an iϋ-group. (See Theorem 2.) Using this condition, we can prove, for example, that the group of the figure eight knot is an i2-group. (See Proposition 3 or Proposition 5.)