Parabolic Representations of Knot Groups, I
Parabolic Representations of Knot Groups, I
复制标题
结群的抛物线表示，I
DOI：
10.1112/plms/s3-24.2.217
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发表时间：
1972
影响因子：
1.8
通讯作者：
Robert F. Riley
中科院分区：
文献类型：
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作者：
Robert F. Riley
设TTK为结群，设6：TTK->PSL(2,.F)为TTK到域F上的2阶矩阵特殊射影群的同态。设x e TTK为子午元，并考虑x在6下的图像xd。如果xd是PSL(2,.F)的抛物线元素，并且如果6的图像是PSL(2,.F)的非阿贝尔子群。 F)，我们将6称为TTK的抛物线表示。很容易看出，如果 xd 对于一个子午线来说是抛物线形，那么对于每个子午线来说也是如此，并且 TTK 的过度呈现的过度生成器是一条子午线。因此，第一个条件 9 是抛物线可以方便地用过生成元来表示。一个非常重要的特殊情况是，其中 F 是具有 p 个元素的素数域，即 p 个素数，因此 L^= PSL (2, _p) 是一个有限群，如果 p# 2, p^ 3 则很简单。Lp 的抛物线元素是 p 阶的元素，并且可以证明抛物线表示 6：TTK-> Lp 必须在（参见下面的第 6 节）。在早期的论文 ([11]) 中，我们表明可以通过一系列易于编程并由计算机执行的实验来找到 -nK 到 Lp 的抛物线表示。这些实验针对数百个结和各种素数 _jp^ 31 进行。我们的第一个主要发现是，对于我们的测试结，群唇上的抛物线表示非常多，并且具有亚历山大多项式 A (x)= 1 的结的抛物线表示似乎并不比其他结少。其次，我们发现与群 Jjp 上的抛物线表示相关的同源不变量是区分结类型的方便且有效的工具。这些经验结果需要解释，在本文中，我们将通过证明如果 K 是 2 桥结类型或环面结类型，则 TTK 在 PSL (2, C) 中具有抛物线表示，其中 C 是复数域。由此可知，TTK 对每个特征都有抛物线表示，其中同态 6 的特征：TTK-»-PSL (2, F) 我们指的是域 F 的特征。作为经典数论的直接结果，我们发现对于无穷多个素数 p，TTK 在 hp 上具有抛物线表示。
Let TTK be a knot group, and let 6: TTK-> PSL (2,. F) be a homomorphism of TTK into the special projective group of rank 2 matrices over a field F. Let x e TTK be a meridian element, and consider the image xd of x under 6. If xd is a parabolic element of PSL (2,. F), and if the image of 6 is a nonabelian subgroup of PSL (2,. F), we shall call 6 a parabolic representation of TTK. It is easy to see that if xd is parabolic for one meridian the same will be true for every meridian, and that an over generator of an over presentation for TTK is a meridian. Hence the first condition that 9 be parabolic can be conveniently stated in terms of over generators. A very important special case is where F is the prime field with p elements, p prime, so that L^= PSL (2, _p) is a finite group which is simple if p# 2, p^ 3. The parabolic elements of Lp are those of order p, and it can be shown that a parabolic representation 6: TTK-> Lp must be onto (cf. § 6 below). In an earlier paper ([11]) we showed that the parabolic representations of-nK to Lp can be found by a sequence of experiments that is easily programmed for execution by a computer. These experiments were carried out for several hundred knots and for various primes _jp^ 31. Our first main discovery was that parabolic representations on the groups lip are remarkably numerous for our test knots, and that parabolic representations for knots having Alexander polynomial A (x)= 1 do not appear to be scarcer than for other knots. Secondly, we found that the homology invariants associated with a parabolic representation on a group Jjp are convenient and effective tools for distinguishing knot type. These empirical results need to be explained, and in this paper we shall begin the explanation by proving that, if K is either the type of a 2-bridge knot or the type of a torus knot, then TTK has parabolic representations in PSL (2, C), where C is the field of complex numbers. It will follow from this that TTK has parabolic representations of every characteristic, where by the characteristic of a homomorphism 6: TTK-»-PSL (2, F) we mean the characteristic of the field F. As an immediate consequence of classical number theory we find that TTK has a parabolic representation on hp for infinitely many primes p.