A rational noncommutative invariant of boundary links
A rational noncommutative invariant of boundary links
复制标题
边界链接的有理非交换不变量
DOI：
10.2140/gt.2004.8.115
复制
发表时间：
2001
影响因子：
2
通讯作者：
A. Kricker
中科院分区：
文献类型：
--
作者：
S. Garoufalidis;A. Kricker
1999年，Rozansky猜想纽结的Kontsevich积分存在有理表示。粗略地说，康采维奇积分的这种有理表示将形式的幂函数级数求和成具有规定分母的有理函数。罗赞斯基的猜想很快就被第二作者证明了。我们首先回顾罗赞斯基的猜想和导致其证明的主要思想。将这一猜想推广到链环的自然问题是边界链环的类，并在这种情况下证明了Rozansky的猜想。一个微妙的问题是，用头发的指数代替珠子的头发贴图并不是1-1。这就提出了一个问题，即边界环的有理不变量是否存在于三叶图的适当空间中，这些图的边被非交换变量中的有理函数装饰。本文的一个主要结果是利用所谓的边界链外科观点，在发展了一种形式化的图式高斯积分之后，构造了这样一个不变量。由于我们的不变量是Kontsevich积分的许多有理形式之一，人们可能会问，我们的不变量是否在某种意义上是正则的。我们通过公理地将我们的不变量刻画为边界环关于零点移动的泛有限型不变量，从而证明了这一点。最后，我们讨论了有理不变量与同调运算之间的关系，并给出了它在低维拓扑中的一些应用。
In 1999, Rozansky conjectured the existence of a rational presentation of the Kontsevich integral of a knot. Roughly speaking, this rational presentation of the Kontsevich integral would sum formal power series into rational functions with prescribed denominators. Rozansky's conjecture was soon proven by the second author. We begin our paper by reviewing Rozansky's conjecture and the main ideas that lead to its proof. The natural question of extending this conjecture to links leads to the class of boundary links, and a proof of Rozansky's conjecture in this case. A subtle issue is the fact that a 'hair' map which replaces beads by the exponential of hair is not 1-1. This raises the question of whether a rational invariant of boundary links exists in an appropriate space of trivalent graphs whose edges are decorated by rational functions in noncommuting variables. A main result of the paper is to construct such an invariant, using the so-called surgery view of boundary links and after developing a formal diagrammatic Gaussian integration. Since our invariant is one of many rational forms of the Kontsevich integral, one may ask if our invariant is in some sense canonical. We prove that this is indeed the case, by axiomatically characterizing our invariant as a universal finite type invariant of boundary links with respect to the null move. Finally, we discuss relations between our rational invariant and homology surgery, and give some applications to low dimensional topology.
曲结理论
DOI：
--
发表时间：
2012
期刊：
影响因子：
--
作者：
K.Iida;N.J.Suematsu;Y.Miyahara;H.Kitahata;M.Nagayama and S.Nakata;鎌田 聖一
通讯作者：
鎌田 聖一