A formula for Casson’s invariant
A formula for Casson’s invariant
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卡森不变量的公式
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发表时间：
1986
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通讯作者：
J. Hoste
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作者：
J. Hoste
设H是一个同调3-球面，它是在同调3-球面M中的一个链环L上通过Dehn手术得到的。若L的每对分支在M中的联结数为零，则我们给出了Casson不变量X（H）的公式，它表示为A（M），L的外科系数，以及L的每个Conway多项式及其所有子联结的系数.给出了这个公式的几个推论。导论. Andrew Casson发现了定向同调3-球面的一个积分不变量，它在模2下约化为Rochlin或n-不变量[1]。本文给出了在同调球面M中，通过对环L的Dehn手术得到同调球面H，且L的每对分支之间的环数为零的情况下的Casson不变量的一个公式。(Note每个同调3-球面都可以用这种方法得到，M = S3。）在这种情况下，卡森不变量X（H）可以用X（M）、L的外科手术系数和来自L及其所有子链的每个康威多项式的某个系数来表示。为了说明这个公式的实用性，我们计算的卡森不变的同源球出现的循环分支覆盖分支沿着无扭双结。我们还描述了如何X-不变的变化时，交叉的框架链接L（两股相同的组件之间）被改变。至少在这种情况下，这提供了一种计算不变量的有效方法。一个有趣的问题是，是否有可能直接证明我们给出的公式被Kirby-Rolfsen演算保持，从而给出X不变性的另一种证明。如果L1和L2是表示同一流形的两个（有理）框架链接，则L1可以通过三种类型的移动序列转换为L2：“同胚”，“扭曲”和“平凡插入或删除”[8]。如果L1的每一对分量之间的链接数为零，则前两种移动类型将保持此属性，而第三种移动可能不会。然而，如果L1可以通过一系列的链接转换成L2，所有的链接数都是零，然后我们证明，X的公式是保留的。我们不知道这样的转变在一般情况下是否可能。为了说明卡森定理和X（H）的公式，我们首先建立一些符号。假设L = { K1，...，Kn }是3-流形M中的一个框架链接，由编辑于1985年10月22日接收。1980年数学学科分类（1985年修订）。第57 M25“由NSF在资助号DMS-8502940下部分支持。X）1986年美国数学学会0002-9947/86 $1.00 + $.25每页
Suppose H is a homology 3-sphere obtained by Dehn surgery on a link L in a homology 3-sphere M. If every pair of components of L has zero linking number in M, then we give a formula for the Casson invariant, X(H), in terms of A (M), the surgery coefficients of L, and a certain coefficient from each of the Conway polynomials of L and all its sublinks. A few consequences of this formula are given. Introduction. Andrew Casson has discovered an integral invariant of oriented homology 3-spheres which reduces, mod 2, to the Rochlin, or n-invariant [1]. In this paper we give a formula for Casson's invariant in the case where the homology sphere H is obtained by Dehn surgery on a link L in a homology sphere M, and furthermore the linking number between every pair of components of L is zero. (Note that every homology 3-sphere can be obtained in this way with M = S3.) In this case the Casson invariant, X(H), can be expressed in terms of X(M), the surgery coefficients of L, and a certain coefficient from each of the Conway polynomials of L and all its sublinks. To illustrate the utility of this formula, we compute the Casson invariant of homology spheres that arise as cyclic branched covers branched along untwisted double knots. We also describe how the X-invariant changes when a crossing of the framed link L (between two strands of the same component) is changed. This provides, at least in this setting, an effective method for computing the invariant. An interesting question is whether it is possible to show directly that the formula we give is preserved by the Kirby-Rolfsen calculus and thus give an alternative proof of the invariance of X. If L1 and L2 are two (rationally) framed links representing the same manifold, then L1 can be transformed into L2 by a sequence of three types of moves: "homeomorphism," "twisting," and "trivial insertion or deletion" [8]. If the linking number between every pair of components of L1 is zero, then the first two types of moves will preserve this property while the third move may not. However, if L1 can be transformed into L2 through a sequence of links all of whose linking numbers are all zero, then we show that the formula for X is preserved. We do not know if such a transformation is, in general, possible. In order to state Casson's theorem and the formula for X(H) we first establish some notation. Suppose L = { K1,..., Kn } is a framed link in a 3-manifold M with Received by the editors October 22, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 57M25. 'Supported in part by the NSF under Grant No. DMS-8502940. X)1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page