Interior polynomial for signed bipartite graphs and the HOMFLY polynomial
Interior polynomial for signed bipartite graphs and the HOMFLY polynomial
复制标题
有符号二分图的内部多项式和 HOMFLY 多项式
DOI：
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发表时间：
2017
影响因子：
0.5
通讯作者：
Keiju Kato
中科院分区：
文献类型：
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作者：
Keiju Kato
内部多项式是二部图的Tutte型不变量，一种特殊的交错链环的HOMFLY多项式的一部分与该链环的Seifert图的内部多项式重合。我们扩展的内部多项式签署二部图，我们表明，在平面的情况下，它等于最大的[公式：见文字]度部分的HOMFLY多项式的一个自然相关的链接。请注意，后者可以是任何定向链接。这个结果适合于一个程序，旨在从Floer同源性推导HOMFLY多项式。我们还建立了一些其他的，更基本的性质的签署内部多项式。例如，[Formula：see text]的镜像的HOMFLY多项式由[Formula：see text]给出。这意味着在平面的情况下，有符号的内部多项式的镜像公式。我们证明了同样的性质适用于任何二部图和相同的图与所有的符号反转。证明依赖于应用于所谓的根多面体的Ehrhart互易性。我们还建立了公式的启发，结理论概念的flyping和突变的签署内部多项式。这导致新的身份为原来的无符号内部多项式。
The interior polynomial is a Tutte-type invariant of bipartite graphs, and a part of the HOMFLY polynomial of a special alternating link coincides with the interior polynomial of the Seifert graph of the link. We extend the interior polynomial to signed bipartite graphs, and we show that, in the planar case, it is equal to the maximal [Formula: see text]-degree part of the HOMFLY polynomial of a naturally associated link. Note that the latter can be any oriented link. This result fits into a program aimed at deriving the HOMFLY polynomial from Floer homology. We also establish some other, more basic properties of the signed interior polynomial. For example, the HOMFLY polynomial of the mirror image of [Formula: see text] is given by [Formula: see text]. This implies a mirroring formula for the signed interior polynomial in the planar case. We prove that the same property holds for any bipartite graph and the same graph with all signs reversed. The proof relies on Ehrhart reciprocity applied to the so-called root polytope. We also establish formulas for the signed interior polynomial inspired by the knot theoretical notions of flyping and mutation. This leads to new identities for the original unsigned interior polynomial.