Surgery formulae for finite type invariants of rational homology 3–spheres
Surgery formulae for finite type invariants of rational homology 3–spheres
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有理同调 3-球体有限型不变量的手术公式
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发表时间：
2007
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通讯作者：
C. Lescop
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作者：
C. Lescop
首先，我们给出了有理同调球面的KontsevichKuperberg-Thurston通用有限型不变量的n次部分Zn的四个图形运算公式。这四个公式中的每一个都确定了形式为X I 1/2N（i1）]I Zn（MI）的交替和，其中N是在有理同调球面M上执行的不相交运算的有限集合，MI表示由I中的运算产生的流形。第一个公式处理的情况下，当N是一组2n拉格朗日保持手术（拉格朗日保持手术取代有理同调体由另一个这样的，而不改变其外部的连接数的曲线）。在第二个公式中，N是边界链路的组件上的n个Dehn手术的集合。第三个公式处理的情况下，3 n外科手术的组件代数分裂链接。第四个公式是2n次手术的组件代数分裂链接，其中所有米尔诺三重链接数消失。在同调球的情况下，这些公式可以被看作是Garoufalidis-Goussarov-Polyak比较的一个改进，该比较是由定向同调球自由生成的有理向量空间的不同滤子（直到定向保持同胚）。然后将所提出的公式应用于研究纽结K在p/q手术下Zn的变化。当Q/Z中的q/p的类固定时，这种变分是q/p中的n次多项式，这些多项式的系数是纽结不变量，对于这些系数给出了各种拓扑性质或拓扑定义.
We first present four graphic surgery formulae for the degree n part Zn of the KontsevichKuperberg-Thurston universal finite type invariant of rational homology spheres. Each of these four formulae determines an alternate sum of the form X I½N (i1) ]I Zn(MI) where N is a finite set of disjoint operations to be performed on a rational homology sphere M, and MI denotes the manifold resulting from the operations in I. The first formula treats the case when N is a set of 2n Lagrangian-preserving surgeries (a Lagrangian-preserving surgery replaces a rational homology handlebody by another such without changing the linking numbers of curves in its exterior). In the second formula, N is a set of n Dehn surgeries on the components of a boundary link. The third formula deals with the case of 3n surgeries on the components of an algebraically split link. The fourth formula is for 2n surgeries on the components of an algebraically split link in which all Milnor triple linking numbers vanish. In the case of homology spheres, these formulae can be seen as a refinement of the Garoufalidis-Goussarov-Polyak comparison of different filtrations of the rational vector space freely generated by oriented homology spheres (up to orientationpreserving homeomorphisms). The presented formulae are then applied to the study of the variation of Zn under a p/qsurgery on a knot K. This variation is a degree n polynomial in q/p when the class of q/p in Q/Z is fixed, and the coefficients of these polynomials are knot invariants, f or which various topological properties or topological definitions are given.