Knot colouring polynomials
Knot colouring polynomials
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结着色多项式
DOI：
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发表时间：
2007
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影响因子：
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通讯作者：
Michael Eisermann
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文献类型：
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作者：
Michael Eisermann
本文介绍了纽结着色数的一种自然推广，称为着色多项式，并研究了它们与杨-巴克斯特不变量和Quandle 2-上循环不变量的关系。
对于3-球面中的纽结K，设pi_K是纽结补的基本群，（m_K，l_K）是pi_K中的一个经纬度对.给定有限群G和G中的元素x，我们考虑表示集
将子午线m_K映射到x，并将着色多项式P（K）定义为所有经度图像上的和
ho（l_K）.由此产生的不变量将节点映射到群环Z[G]。它对于连通和是乘法的，对于纽结的对称运算是等变的。给出的例子表明，着色多项式可以区分其他不变量无法区分的结，特别是它们可以区分结与其突变体、正、逆或逆。
证明了纽结的每一个Quandle 2-上循环状态和不变量都是某个纽结着色多项式的特殊化。这提供了一个完整的拓扑解释这些不变量的纽结组及其周边系统。此外，我们证明了P可以表示为杨-巴克斯特不变量，即作为一些线性辫子群表示的迹。这特别需要杨-巴克斯特不变量可以检测不可逆和不可逆的结。
This article introduces a natural extension of colouring numbers of knots, called colouring polynomials, and studies their relationship to Yang-Baxter invariants and quandle 2-cocycle invariants.
For a knot K in the 3-sphere let pi_K be the fundamental group of the knot complement, and let (m_K,l_K) be a meridian-longitude pair in pi_K. Given a finite group G and an element x in G, we consider the set of representations
ho from pi_K to G that map the meridian m_K to x, and define the colouring polynomial P(K) as the sum over all longitude images
ho(l_K). The resulting invariant maps knots to the group ring Z[G]. It is multiplicative with respect to connected sum and equivariant with respect to symmetry operations of knots. Examples are given to show that colouring polynomials distinguish knots for which other invariants fail, in particular they can distinguish knots from their mutants, obverses, inverses, or reverses.
We prove that every quandle 2-cocycle state-sum invariant of knots is a specialization of some knot colouring polynomial. This provides a complete topological interpretation of these invariants in terms of the knot group and its peripheral system. Furthermore, we show that P can be presented as a Yang-Baxter invariant, i.e. as the trace of some linear braid group representation. This entails in particular that Yang-Baxter invariants can detect non-inversible and non-reversible knots.
影响因子：
0.7
作者：
.Ilker S. Yuce-Ilker-S.-Yuce-102800584
通讯作者：
.Ilker S. Yuce-Ilker-S.-Yuce-102800584