Bottom tangles and universal invariants
Bottom tangles and universal invariants
复制标题
DOI：
10.2140/agt.2006.6.1113
复制
发表时间：
2005-05
影响因子：
0.7
通讯作者：
K. Habiro
中科院分区：
文献类型：
--
作者：
K. Habiro
底部缠结是立方体内的一种缠结，它仅由弧组件组成，每个弧组件的两个端点都在立方体的底线上，且彼此相邻放置。我们引入了有框定向缠结范畴的一个子范畴\(B\)，它作用于底部缠结的集合。我们给出了\(B\)的一组有限生成元，这为生成所有底部缠结，进而通过闭包生成所有有框定向链环提供了一种特别便捷的方式。我们还在底部缠结的集合上定义了一种“辫子霍普夫代数作用”。利用与每个带状霍普夫代数\(H\)相关联的底部缠结的通用不变量，我们定义了一个从\(B\)到左\(H\)-模范畴\(\text{Mod}_H\)的辫子函子\(J\)。函子\(J\)以及\(B\)的生成元集合为研究链环的量子不变量的范围提供了一种代数方法。底部缠结上的辫子霍普夫代数作用通过\(J\)映射到\(\text{Mod}_H\)中\(H\)的标准辫子霍普夫代数结构。纽结理论中的几个概念，如亏格、解结数、带状纽结、边界链环、局部变换等，在涉及范畴\(B\)的设定中都被赋予了代数解释。函子\(J\)为研究这些概念与量子不变量之间的关系提供了一种便捷的方式。
57M27；57M25，18D10（这部分可能是数学文献分类号之类的内容，未做详细翻译处理，如果有特殊要求可进一步说明）
A bottom tangle is a tangle in a cube consisting only of arc components, each of which has the two endpoints on the bottom line of the cube, placed next to each other. We introduce a subcategory B of the category of framed, oriented tangles, which acts on the set of bottom tangles. We give a finite set of generators of B, which provides an especially convenient way to generate all the bottom tangles, and hence all the framed, oriented links, via closure. We also define a kind of “braided Hopf algebra action” on the set of bottom tangles. Using the universal invariant of bottom tangles associated to each ribbon Hopf algebra H , we define a braided functor J from B to the category ModH of left H ‐modules. The functor J, together with the set of generators of B, provides an algebraic method to study the range of quantum invariants of links. The braided Hopf algebra action on bottom tangles is mapped by J to the standard braided Hopf algebra structure for H in ModH . Several notions in knot theory, such as genus, unknotting number, ribbon knots, boundary links, local moves, etc are given algebraic interpretations in the setting involving the category B. The functor J provides a convenient way to study the relationships between these notions and quantum invariants. 57M27; 57M25, 18D10