Isotropic systems and the interlace polynomial

TL;DR

This paper demonstrates how isotropic systems can be used to derive a closed-form expression for the vertex-nullity interlace polynomial, highlighting their connection to the Tutte-Martin polynomial and offering new insights into graph invariants.

Contribution

It provides an alternative proof for the interlace polynomial's closed form using isotropic systems, illustrating their utility in graph polynomial analysis.

Findings

Connected the interlace polynomial to the Tutte-Martin polynomial of isotropic systems.

Presented a new proof for the polynomial's closed form.

Showcased isotropic systems as a tool for investigating graph invariants.

Abstract

Through a series of papers in the 1980's, Bouchet introduced isotropic systems and the Tutte-Martin polynomial of an isotropic system. Then, Arratia, Bollob\'as, 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, Bollob\'as, 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.