论文标题
通过虚拟链接定义边界的链接和3个manifolds的量子不变性：计算某些示例
Quantum Invariants of Links and 3-Manifolds with Boundary defined via Virtual Links: Calculation of some examples
论文作者
论文摘要
在本文的前传中，考夫曼（Kauffman）和奥加萨（Ogasa）引入了与边界的紧凑定向的3个manifolds的新拓扑量子不变式，其中边界是两个相同表面的不相交联合。不变性是通过手术构建的，以$ f \ times i $的流形为单位，其中$ i $表示单位间隔。由于虚拟结和链接被表示为如此厚的表面中的链接，因此我们还可以根据虚拟链接图（带虚拟交叉的平面图）来构造不变性。 这些不变的是具有非呈现边界的3个manifolds的量子不变式的新的，非平凡的和可计算的示例。 由于虚拟结和链接由圆形表面中的圆圈的嵌入表示，因此我们将3键中的圆圈嵌入为{\ it经典链接}。经典链接与可以在增厚的2个球体中表示的虚拟链接相同，这是一个事实，即嵌入了同位素的虚拟链接的集合中，经典链接嵌入了同位素。我们从以下意义上讲，我们在三个角度上给出了一个新的经典链接的不变性：考虑两个组件的$ s^3 $中的链接$ l $。 $ l $的管状社区的补充是一种多种多样的边界，由两个圆环的副本组成。我们的不变性适用于这种有界的歧管，并提供两个组件的给定链接的新不变。还获得了结的不变。 在本文中，我们明确计算了一些示例的拓扑量子不变性。我们从示例中得出结论，我们的不变是新的，足够强大，足以区分一些古典结。 我们还解释了三个manifolds的拓扑量子不变式与边界和Reshetikhin-Turaev不变式是如何不同的。 （有关细节，请参见身体）。
In the prequel of this paper, Kauffman and Ogasa introduced new topological quantum invariants of compact oriented 3-manifolds with boundary where the boundary is a disjoint union of two identical surfaces. The invariants are constructed via surgery on manifolds of the form $F \times I$ where $I$ denotes the unit interval. Since virtual knots and links are represented as links in such thickened surfaces, we are able also to construct invariants in terms of virtual link diagrams (planar diagrams with virtual crossings). These invariants are new, nontrivial, and calculable examples of quantum invariants of 3-manifolds with non-vacuous boundary. Since virtual knots and links are represented by embeddings of circles in thickened surfaces, we refer to embeddings of circles in the 3-sphere as {\it classical links}. Classical links are the same as virtual links that can be represented in a thickened 2-sphere and it is a fact that classical links, up to isotopy, embed in the collection of virtual links taken up to isotopy. We give a new invariant of classical links in the 3-sphere in the following sense: Consider a link $L$ in $S^3$ of two components. The complement of a tubular neighborhood of $L$ is a manifold whose boundary consists in two copies of a torus. Our invariants apply to this case of bounded manifold and give new invariants of the given link of two components. Invariants of knots are also obtained. In this paper we calculate the topological quantum invariants of some examples explicitly. We conclude from our examples that our invariant is new and strong enough to distinguish some classical knots from one another. We also explain how different Our topological quantum invariants of 3-manifolds with boundary and the Reshetikhin-Turaev invariants are. (See the body for detail).
