arXiv 1602.061912016-02-19math.GT

带状缠结与平面代数的Alexander不变量

Alexander invariants of ribbon tangles and planar algebras

Celeste Damiani, Vincent Florens

摘要：

带状缠结是圆环面和圆柱面在$4$ -球~ $B^4$中的适当嵌入，“绑定”$3$ -流形只有带状盘作为奇点。我们构造了一个亚历山大不变量$\mathsf{A}$的带状缠结配备了一个表示的基本群的外部在一个自由阿尔贝群$G$。这个不变量在缠结的某个范畴$\mathsf{R}ib_G$中诱导了一个函子，它限制了带状编织物的Burau-Gassner表示的外幂，这在本文中类似于通常的编织物。我们定义了一个电路代数$\mathsf{C}ob_G$上的操作的光滑协边，启发由琼斯介绍的图解平面代数，并证明了不变量$\mathsf{A}$与该代数的组成交换。另一方面，丝带缠结承认图解表示，正确焊接图。我们给出了一个简单的组合描述$\mathsf{A}$和代数$\mathsf{C}ob_G$，并观察到，我们的建设是一个拓扑化身的亚历山大不变量的阿奇博尔德。当被限制在没有虚拟交叉的图中时，$\mathsf{A}$提供了一个对通常的亚历山大多项式的纯粹局部描述，并扩展了毕格罗、卡塔布里加和第二作者的构造。

英文摘要：

Ribbon tangles are proper embeddings of tori and cylinders in the $4$-ball~$B^4$, "bounding" $3$-manifolds with only ribbon disks as singularities. We construct an Alexander invariant $\mathsf{A}$ of ribbon tangles equipped with a representation of the fundamental group of their exterior in a free abelian group $G$. This invariant induces a functor in a certain category $\mathsf{R}ib_G$ of tangles, which restricts to the exterior powers of Burau-Gassner representation for ribbon braids, that are analogous to usual braids in this context. We define a circuit algebra $\mathsf{C}ob_G$ over the operad of smooth cobordisms, inspired by diagrammatic planar algebras introduced by Jones, and prove that the invariant $\mathsf{A}$ commutes with the compositions in this algebra. On the other hand, ribbon tangles admit diagrammatic representations, throught welded diagrams. We give a simple combinatorial description of $\mathsf{A}$ and of the algebra $\mathsf{C}ob_G$, and observe that our construction is a topological incarnation of the Alexander invariant of Archibald. When restricted to diagrams without virtual crossings, $\mathsf{A}$ provides a purely local description of the usual Alexander poynomial of links, and extends the construction by Bigelow, Cattabriga and the second author.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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