标题： 纽结族的Vassiliev不变量的代数结构 显示英文标题

标题： Algebraic structures of Vassiliev invariants for knot families

摘要： 我们探讨通过3维Chern-Simons理论中的关联函数表达的Vassiliev纽结不变量之间的代数关系。 Vassiliev不变量形成一个无限维代数。 我们关注$k$参数化的纽结族，其中Vassiliev不变量是族参数的多项式。 结果表明，这样的1参数化的Vassiliev不变量代数总是有限生成的，而在更多参数的情况下，生成元的数量可能是无限的。 在纽结族内部，会出现关于Vassiliev不变量的额外代数关系。 我们证明对于$k$参数化的纽结族，存在$\leq k$个代数独立的Vassiliev不变量。 然而，在我们所有的例子中，代数独立的Vassiliev不变量的数量正好是$k$，并且这是一个开放问题，即是否存在一个$k$参数化的纽结族，其代数独立的Vassiliev不变量数量更少。 我们还证明了一些$k$参数化的纽结族的完整纽结不变量由$\leq k$个Vassiliev不变量组成。 再次，我们只有$k$个 Vassiliev 不变量构成的集合是$k$参数扭结族的完整不变量的例子。 目前，尚不清楚更少数量的 Vassiliev 不变量的集合是否不能成为扭结族的完整不变量。 显示英文摘要

摘要： We explore algebraic relations on Vassiliev knot invariants expressed through correlators in the 3-dimensional Chern-Simons theory. Vassiliev invariants form an infinite-dimensional algebra. We focus on $k$-parametric knot families with Vassiliev invariants being polynomials in family parameters. It turns out that such a 1-parametric algebra of Vassiliev invariants is always finitely generated, while in the case of more parameters, the number of generators can be infinite. Inside a knot family, there appear extra algebraic relations on Vassiliev invariants. We show that there are $\leq k$ algebraically independent Vassiliev invariants for a $k$-parametric knot family. However, in all our examples, the number of algebraically independent Vassiliev invariants is exactly $k$, and it is an open question if there exists a $k$-parametric knot family with a fewer number of algebraically independent Vassiliev invariants. We also demonstrate that a complete knot invariant of some $k$-parametric knot families consists of $\leq k$ Vassiliev invariants. Again, we have only examples of a set of $k$ Vassiliev invariants being a complete invariant of a $k$-parametric knot family. Currently, it is unknown whether a set of a fewer number of Vassiliev invariants cannot be a complete knot family invariant.