Poincar\'e polynomial of elliptic arrangements is not a specialization of the Tutte polynomial

TL;DR

This paper demonstrates that the Poincaré polynomial of certain elliptic arrangements cannot be derived as a specialization of the Tutte polynomial, highlighting limitations in the relationship between these invariants.

Contribution

It provides explicit examples of elliptic arrangements with identical Tutte polynomials but different Betti numbers, challenging previous assumptions about their connection.

Findings

Two elliptic arrangements share the same Tutte polynomial.

These arrangements have different Betti numbers.

The Poincaré polynomial is not a specialization of the Tutte polynomial.

Abstract

The Poincar\'e polynomial of the complement of an arrangements in a non compact group is a specialization of the $G$-Tutte polynomial associated with the arrangement. In this article we show two unimodular elliptic arrangements (built up from two graphs) with the same Tutte polynomial, having different Betti numbers.