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首页> 外文期刊>Journal of knot theory and its ramifications >GRAPH-VALUED INVARIANTS OF VIRTUAL AND CLASSICAL LINKS AND MINIMALITY PROBLEM
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GRAPH-VALUED INVARIANTS OF VIRTUAL AND CLASSICAL LINKS AND MINIMALITY PROBLEM

机译：虚拟和经典链接的图形值不变性和极小问题

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摘要

The Kuperberg bracket is a well-known invariant of classical links. Recently, the second named author and Kauffman constructed the graph-valued generalization of the Kuperberg bracket for the case of virtual links: unlike the classical case, the invariant in the virtual case is valued in graphs which carry a significant amount of information about the virtual knot. The crucial difference between virtual knot theory and classical knot theory is the rich topology of the ambient space for virtual knots. In a paper by Chrisman and the second named author, two-component classical links with one fibered component were considered; the complement to the fibered component allows one to get highly non-trivial ambient topology for the other component. In this paper, we combine the ideas of the above mentioned papers and construct the "virtual" Kuperberg bracket for two-component links L = J ∪ Κ with one component (J) fibered. We consider a new geometrical complexity for such links and establish minimality of diagrams in a strong sense. Roughly speaking, every other "diagram" of the knot in question contains the initial diagram as a subdiagram. We prove a sufficient condition for minimality in a strong sense where minimality cannot be established as introduced in the paper by Chrisman and the second named author.
机译：Kuperberg括号是经典链接的著名不变量。最近，第二名作者和考夫曼（Kauffman）为虚拟链接的情况构造了Kuperberg括号的图值概括：与经典情况不同，在虚拟情况下的不变性在包含大量有关虚拟信息的图形中得到了重视。结。虚拟结理论与经典结理论之间的关键区别在于虚拟结的周围空间的丰富拓扑。在克里斯曼（Chrisman）和第二位命名作者的论文中，考虑了带有一个纤维成分的两成分经典链接。纤维组件的互补性使一个组件能够获得高度平凡的环境拓扑。在本文中，我们结合了上述论文的思想，并为带有纤维的一个成分（J）的两成分链接L = J∪Κ构造了“虚拟” Kuperberg托架。我们考虑了此类链接的新几何复杂性，并从强烈的意义上建立了图表的最小化。粗略地说，所讨论的结的每个其他“图”都包含初始图作为子图。正如克里斯曼和第二名作者在论文中所介绍的那样，在很强的意义上我们无法证明极小化的条件下证明了极小化的充分条件。

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