Polynomiality threshold for graphs of higher circumference

Determine whether, for graphs of circumference 2, the Betti number sequence \(\dim_F H_i(B_k\Gamma)\) becomes polynomial at braid index \(k=1\) or \(k=2\), and determine the corresponding onset of polynomiality for graphs whose circumference exceeds 2.

Background

The paper establishes exact Hilbert-polynomial formulas for bunches of grapes, namely graphs of topological circumference at most 1. It observes that graphs of circumference 2 can be obtained by replacing stem edges with multiple edges, but the multiple-edge action prevents a direct extension of the injection used in the circumference-one case. Known special cases give different stabilization thresholds, motivating the unresolved question of whether the threshold is generally 1 or 2 and what happens for larger circumference.

References

For a graph \Gamma with circumference 2, does the polynomiality of the Betti number \dim_F H_i(B_k\Gamma) begin at k=1 or 2? What if the circumference of \Gamma is larger than 2?

— Hilbert polynomials of configuration spaces over graphs of circumference at most 1  (2505.24416 - An et al., 30 May 2025) in Section 1, subsection “Future directions,” item 2