各向同性系统与交错多项式
Isotropic systems and the interlace polynomial
摘要：
在1980年代的一系列论文中，Bouchet介绍了各向同性系统和各向同性系统的Tutte-Martin多项式。然后，Arratia，Bollobás和Sorkin在[ABS 00]中开发了一个图形的交错多项式，以响应DNA测序应用。交错多项式最近引起了相当大的关注，新的结果包括通过封闭形式的生成函数表达式而不是原始递归定义来实现原始交错多项式（参见Aigner和van der Holst [AvdH 04]，以及Arratia，Bollobás和Sorkin [ABS 04 b]）。现在，Bouchet [Bou 05]将图的顶点零交错多项式识别为相关联的各向同性系统的Tutte-Martin多项式。这表明各向同性系统的机制可能非常适合于研究交错多项式的性质。因此，我们在这里提出了一个替代的证明的封闭形式表示的顶点零交错多项式使用的机器的各向同性系统。这种方法既说明了顶点零交错多项式与各向同性系统的Tutte-Martin多项式之间的密切联系，也提供了一个操纵各向同性系统的具体例子。我们还简要介绍了相关工作。
英文摘要：
Through a series of papers in the 1980's, Bouchet introduced isotropic systems and the Tutte-Martin polynomial of an isotropic system. Then, Arratia, Bollobás, and Sorkin developed the interlace polynomial of a graph in [ABS00] in response to a DNA sequencing application. The interlace polynomial has generated considerable recent attention, with new results including realizing the original interlace polynomial by a closed form generating function expression instead of by the original recursive definition (see Aigner and van der Holst [AvdH04], and Arratia, Bollobás, and Sorkin [ABS04b]). Now, Bouchet [Bou05] recognizes the vertex-nullity interlace polynomial of a graph as the Tutte-Martin polynomial of an associated isotropic system. This suggests that the machinery of isotropic systems may be well-suited to investigating properties of the interlace polynomial. Thus, we present here an alternative proof for the closed form presentation of the vertex-nullity interlace polynomial using the machinery of isotropic systems. This approach both illustrates the intimate connection between the vertex-nullity interlace polynomial and the Tutte-Martin polynomial of an isotropic system and also provides a concrete example of manipulating isotropic systems. We also provide a brief survey of related work.
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