# On the Falk invariant of signed graphic arrangements

## TL;DR

This paper provides a combinatorial formula for the Falk invariant of signed graphic arrangements, enhancing understanding of their topological properties by linking algebraic invariants to combinatorial structures.

## Contribution

It introduces a new combinatorial formula for the Falk invariant applicable to signed graphic arrangements excluding those with a B2 sub-arrangement.

## Key findings

- Derived a combinatorial formula for the Falk invariant
- Applied the formula to signed graphic arrangements without B2 sub-arrangements
- Enhanced understanding of the topological invariants of hyperplane arrangements

## Abstract

The fundamental group of the complement of a hyperplane arrangement in a complex vector space is an important topological invariant. The third rank of successive quotients in the lower central series of the fundamental group was called Falk invariant of the arrangement since Falk gave the first formula and asked to give a combinatorial interpretation. In this article, we give a combinatorial formula for the Falk invariant of a signed graphic arrangement that do not have a $B_2$ as sub-arrangement.