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.

中文翻译：

---

有符号二部图的内部多项式和 HOMFLY 多项式

内部多项式是二部图的Tutte型不变量，特殊交替链路的HOMFLY多项式的一部分与链路的Seifert图的内部多项式重合。我们将内部多项式扩展到有符号二部图，并且我们表明，在平面情况下，它等于自然关联链接的 HOMFLY 多项式的最大 [公式：见文本]-度部分。请注意，后者可以是任何定向链接。这个结果适合一个旨在从 Floer 同调导出 HOMFLY 多项式的程序。我们还建立了带符号内部多项式的一些其他更基本的性质。例如，[公式：见文]的镜像的HOMFLY多项式由[公式：见文]给出。这意味着平面情况下有符号内部多项式的镜像公式。我们证明了相同的性质适用于任何二分图和所有符号反转的相同图。证明依赖于应用于所谓的根多面体的 Ehrhart 互易性。我们还建立了受飞行和突变的结理论概念启发的有符号内部多项式的公式。这导致原始无符号内部多项式的新恒等式。