Combinatorial k-formality of hyperpolygonal arrangements

Prove that for every positive integer \(n\), the hyperpolygonal arrangement \(\mathcal H_n\) is k-formal for every admissible integer \(k\), namely for all \(1\leq k\leq \operatorname{rank}(\mathcal H_n)\).

Background

The paper establishes that every hyperpolygonal arrangement Hn\mathcal H_n is combinatorially formal. It then recalls the stronger notion of k-formality, defined for integers 1≤k≤rank⁡(Hn)1\leq k\leq \operatorname{rank}(\mathcal H_n), and notes that free arrangements are k-formal for every such kk.

Computational evidence from additional non-free hyperpolygonal arrangements motivates the conjecture that the entire hyperpolygonal family has this stronger formality property, despite not all members being free.

References

Computational evidence for further non-free hyperpolygonal arrangements suggests the following. For any $n \in $, $_n$ is $k$-formal for any $k$.

— Hyperpolygonal arrangements  (2502.02274 - Giordani et al., 4 Feb 2025) in Conjecture \ref{conjecture} in Section 5, immediately after the proof of Theorem \ref{thm:HAformal}