Bi-oriented Quantum Algebras, and a Generalized Alexander Polynomial for Virtual Links
Bi-oriented Quantum Algebras, and a Generalized Alexander Polynomial for Virtual Links
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双向量子代数和虚拟链路的广义亚历山大多项式
DOI：
10.1090/conm/318/05548
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发表时间：
2001
期刊：
影响因子：
--
通讯作者：
D. Radford
中科院分区：
文献类型：
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作者：
L. Kauffman;D. Radford
本文讨论了虚拟结和链接的广义亚历山大多项式的构造，以及将该不变量重新表述为量子链接不变量。然后我们引入双向量子代数的概念，它为该结构提供了代数背景。在本文中，我们讨论通过双分方程的概念构造广义亚历山大多项式 GK(s;t)。我们的方法直接导致了这个不变量所基于的 Burau 表示的概括。然后，我们将不变量重新表述为量子链接不变量和状态求和。在这种情况下，我们证明归一化量子不变量 ZK(ae;? ) 满足康威绞丝恒等式并再现 GK(s;t)。这些不变量对于虚拟结和链接的理论很有用，因为它们在经典结和链接上消失。因此，这里研究的不变量可以用来表明许多虚拟结和链接不是经典的。我们在第 2 节中给出了这样的例子。我们还给出了一个无法被广义亚历山大多项式检测到的虚拟结的例子。该结由相应的广义 Alexander 模块的结构检测到。我们以 Kishino 绘制的图表作为结论，该图表是打结的，但在撰写本文时尚未被双分形的任何表示形式检测到。我们推测岸野图的双分形确实检测到了它的结点。在本文的最后部分，我们提出了双向量子代数的概念。这概括了我们之前的定向量子代数 (6, 7) 概念，包括创建虚拟链接不变量所需的结构。本文研究的不变式 ZK(ae;? ) 非常适合这个框架。后续论文将研究双向量子代数的结构和应用。
This paper discusses the construction of a generalized Alexander polynomial for virtual knots and links, and the reformulation of this invariant as a quantum link invariant. We then introduce the concept of a bi-oriented quantum algebra, which provides an algebraic context for this structure. In this paper we discuss the construction of a generalized Alexander polynomial GK(s;t) via the concept of a biquandle. Our approach leads directly to a gener- alization of the Burau representation upon which this invariant is based. We then reformulate the invariant as a quantum link invariant and as a state summation. In this context we show that the normalized quantum invariant ZK(ae;? ) satisfles a Conway skein identity and reproduces GK(s;t). These invariants are useful for the theory of virtual knots and links, for they vanish on classical knots and links. Hence the invariants studied here can be used to show that many virtual knots and links are not classical. We give such examples in Section 2. We also give an example of a virtual knot that cannot be detected by the generalized Alexander polynomial. This knot is detected by the structure of the corresponding generalized Alexander module. We conclude with a diagram due to Kishino that is knotted, but is not detected by any representation of the biquandle at the time of this writing. We conjecture that the biquandle of the Kishino diagram does detect its knottedness. In the flnal section of the paper, we formulate the concept of a bi-oriented quantum algebra. This generalizes our previous concept of oriented quantum alge- bra (6, 7) to include the necessary structures to create invariants of virtual links. The invariant ZK(ae;? ) studied in this paper flts non-trivially into this framework. Subsequent papers will study the structure and applications of bi-oriented quantum algebras.