丝带结组
Groups of ribbon knots
摘要：
给定一个正整数$n$，我们说两个纽结是$V_n$等价的，如果它们具有相同的$\le n$阶的Vassiliev不变量.我们表明，带状结的$V_n$ -等价类形成一个组，操作是由结的连接和诱导。这个群是自由阿贝尔群，它的秩等于$\le n$阶的原始Vassiliev不变量空间的维数。带状结的$n$等价类（遵循Gusarov的概念）也构成一个群。作为推论，任何非带状纽结的Arf不变量是不平凡的，不能通过许多Vassiliev不变量与带状纽结区分开来。此外，所有的加性纽结配边不变量都不是有限型的。作为一个双积，我们证明了对于$n>5$，$n$阶的独立Vassiliev不变量的个数上有界于$(n-2)!/2$.
英文摘要：
Given a positive integer $n$, we say that two knots are $V_n$-equivalent if they have the same Vassiliev invariants of order $\le n$. We showed that the $V_n$-equivalence classes of ribbon knots form a group, the operation being induced by the connected sums of knots. The group is free abelian and its rank is equal to the dimension of the space of primitive Vassiliev invariants of order $\le n$. The $n$-equivalence classes (following Gusarov's notion) of ribbon knots also constitute a group. As a corollary, any non-ribbon knot whose Arf invariant istrivial cannot be distinguished from ribbon knots by finitely many Vassiliev invariants. Furthermore, all additive knot cobordism invariants are not of finite type. As a bi-product, we prove that the number of independent Vassiliev invariants of order $n$ is bounded above by $(n-2)!/2$ for $n>5$.
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