Finite type link invariants of 3-manifolds
Finite type link invariants of 3-manifolds
复制标题
3-流形的有限类型链接不变量
DOI：
10.1016/0040-9383(94)90034-5
复制
发表时间：
1994
期刊：
影响因子：
--
通讯作者：
Xiaoxia Lin
中科院分区：
文献类型：
--
作者：
Xiaoxia Lin
自从琼斯多项式及其推广的发现（例如，参见[6，lo]）以来，人们总是试图理解它们与结和链路拓扑的关系。由于这些连杆不变量的原始定义从根本上依赖于连杆在三维空间中的投影，这当然不是一件容易的事。因此，第一步应该是为这些独立于三维空间的特殊特征的琼斯型链接不变量找到一个定义，从而捕获这些链接不变量的底层拓扑。换句话说，我们希望有一个琼斯型连杆不变量的内在定义，这样它就可以应用于3流形，而不是只受一些拓扑限制的3空间。Witten用chen - simons路径积分定义的jones型不变量[13]，当然可以被认为是这样一个内在定义。尽管路径积分没有严格定义，但我们从chen - simons路径积分的摄动理论[1,21]中得出的教训是，我们应该将jones型连杆不变量视为形式幂级数，其中只有系数可能在几何上或拓扑上定义良好。这些形式幂级数的收敛性没有先验的理由。一些正式的幂级数在三维空间中收敛为HOMFLY和Kauffman多项式因为它们满足一定的交叉变化公式而交叉变化降低了三维空间中连杆的复杂性。然而，由于摄动理论中Chern-Simons路径积分的幂级数展开的系数在一定的有限维辅助流形上被表示为奇异积分，因此存在奇异积分的收敛性以及奇异积分与度量选择的独立性等问题。此外，我们还不清楚从这些奇异积分中可以得到什么样的拓扑信息。
SINCE the discovery of the Jones polynomial and its generalizations(see [6, lo], for example), it is always tempting to try to understand their relations with the topology of knots and links. As the original definition of these link invariants depends in a fundamental way on the projections of links in the 3-space, this is certainly not an easy task. Therefore, the first step should be to find a definition for these Jones-type link invariants independent of the special features of the 3-space and thus to capture the underlying topology of these link invariants. In other words, we would like to have an intrinsic definition of Jones-type link invariants so that it can be applied to 3-manifolds other than the 3-space subject only to some topological restrictions.Witten’s definition of Jones-type invariants [13] in terms of Chern-Simons path integrals could certainly be thought of as such an intrinsic definition. In spite of the fact that path integrals are not rigorously defined, the lesson we drew from the perturbation theory of Chern-Simons path integrals [l, 21 is that we should rather think of Jones-type link invariants as formal power series where only the coefficients might be well-defined geometrically or topologically. There is no reason a prior for the convergence of these formal power series. Some of these formal power series converge to things like HOMFLY and Kauffman polynomials in the 3-space because they satisfy certain crossing change formulae and crossing changes reduce the complexity of links in the 3-space. Nevertheless, as the coefficients of the power series expansions of Chern-Simons path integrals in the perturbation theory are expressed as singular integrations over certain finite dimensional auxiliary manifolds, there are problems like the convergence of these singular integrations and the independence of these singular integrations from the choice of metrics, etc. Moreover, it is not clear what kind of topological information we can obtain from these singular integrations.