A mirroring formula for the interior polynomial of a bipartite graph

TL;DR

This paper proves a mirror symmetry property of the interior polynomial for bipartite graphs, extending known results from planar graphs to all bipartite graphs using Ehrhart reciprocity and root polytopes.

Contribution

It generalizes the mirror property of the interior polynomial from planar bipartite graphs to all bipartite graphs, employing Ehrhart reciprocity and root polytope techniques.

Findings

Interior polynomial of bipartite graphs exhibits mirror symmetry.

Established formulas inspired by knot theory notions like flyping and mutation.

Extended known planar results to all bipartite graphs.

Abstract

The interior polynomial is an invariant of (signed) bipartite graphs, and the interior polynomial of a plane bipartite graph is equal to a part of the HOMFLY polynomial of a naturally associated link. The HOMFLY polynomial $P_L(v,z)$ is a famous link invariant with many known properties. For example, the HOMFLY polynomial of the mirror image of $L$ is given by $P_{L}(-v^{-1},z)$. This implies a property of the interior polynomial in the planar case. We prove that the same property holds for any bipartite graph. The proof relies on Ehrhart reciprocity applied to the so called root polytope. We also establish formulas for the interior polynomial inspired by the knot theoretical notions of flyping and mutation.