Deforming twist-spun knots
Deforming twist-spun knots
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使扭结变形
DOI：
10.1090/s0002-9947-1979-0530058-4
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发表时间：
1979
影响因子：
1.3
通讯作者：
R. Litherland
中科院分区：
文献类型：
--
作者：
R. Litherland
Zeeman在[151]中介绍了将一个n-纽结捻纺成一个（n +1）-纽结的过程，并证明了一个捻纺纽结可解的显著定理。在[21]中，福克斯描述了另一种可以在纺纱过程中应用的变形，他称之为轧制。我们表明，提供一个结合的滚动与扭曲，所产生的结是再次bellared。事实上，这个结果适用于下面定义的更大类的变形。0.导论.纺纱的过程可以描述如下。假设我们在半三维空间R 3 i中有一个打结弧，其端点在R2 = aR 3中。围绕R2旋转R3生成R '，圆弧扫过一个打结的2-球面。现在假设在旋转过程中，我们在R 3中移动弧（保持其端点固定）;如果它在旋转结束时返回到其原始位置，我们再次在R '中获得2-球。我们称这个过程为变形旋转;一个精确的定义（在任何维度）给出了？1（为了方便，我们使用R 3的一点紧化B3）。变形旋转的第一个例子是由塞曼[15]给出的;这是围绕其轴扭转结的操作。随后，福克斯[2]描述了纺纱的另一种变化，称为辊纺纱，并表明这与加捻纺纱不同。（具体来说，他证明了滚纺的8字结不能通过捻纺8字来获得。）现在，[15]的主要结果是一个引人注目的定理，即任何扭转旋转的纽结都是可解的。人们自然会问，这是否可以扩展到滚动，或者滚动和扭转的某种组合。首先，我们需要一个滚动的定义;在[2]福克斯只给出了一个图片滚动一个数字8结，并评论说，该操作类似于“滚动一个袜子”。我们的第一个观察结果是，在福克斯的照片中，弧的点从结的右边移到左边。这意味着，我们应该把绳子穿过结，而不是把结滚到绳子上。换句话说，我们应该考虑一个空间的合痕，它使一个基点围绕纽结移动一次，并且远离纽结固定。不幸的是，这只对打结的圆有意义，我们没有得到相应打结弧的运动，1977年7月14日由编辑收到。AMS（MOS）主题分类（1970年）。初级57 C45，55 A25。
In [151 Zeeman introduced the process of twist-spinning an n-knot to obtain an (n + I)-knot, and proved the remarkable theorem that a twist-spun knot is fibred. In [21 Fox described another deformation which can be applied during the spinning process, and which he called rolling. We show that, provided one combines the rolling with a twist, the resulting knot is again fibred. In fact, this result holds for a larger class of deformations, defined below. 0. Introduction. The process of spinning may be described as follows. Suppose we have a knotted arc in half 3-space R 3i, with its endpoints in R2 = aR3 . Rotating R 3 about R2 generates R', and the arc sweeps out a knotted 2-sphere. Now suppose that during the spinning process we move the arc in R 3 (keeping its endpoints fixed); provided it is returned to its original position at the end of the spinning, we again obtain a 2-sphere in R'. We call this process deform-spinning; a precise definition (in any dimension) is given in ? 1 (where for convenience we work with the one-point compactification B3 of R 3 ). The first example of deform-spinning was given by Zeeman [15]; this was the operation of twisting the knot about its axis. Subsequently, Fox [2] described another variation on spinning, called roll-spinning, and showed that this differs from twist-spinning. (Specifically, he showed that the roll-spun figure eight knot cannot be obtained by twist-spinning the figure eight.) Now, the main result of [15] is the striking theorem that any twist-spun knot is fibred. It is natural to ask whether this can be extended to rolling, or some combination of rolling and twisting. First we need a definition of rolling; in [2] Fox only gives a picture of rolling a figure eight knot, and remarks that the operation resembles "rolling a stocking". Our first observation is that in Fox's pictures points of the arc move from the right of the knot to the left. This suggests that, instead of rolling the knot down the string, we should pull the string through the knot. In other words, we should consider an isotopy of space which moves a base-point once around the knot, and is fixed away from the knot. Unfortunately, this only makes sense for a knotted circle, and we do not get a motion of the corresponding knotted arc, Received by the editors July 14, 1977. AMS (MOS) subject classifications (1970). Primary 57C45, 55A25.