Effectiveness of the q^N_min-based genus bounds

Determine the effectiveness of the Euler characteristic bounds for smoothly embedded oriented surfaces S ⊂ W given by χ(S) ≤ (−N·([S]·[S]) − q^N_min(W,L,[S]))/(N−1), where q^N_min(W,L,α) denotes the minimal q-degree of a nonzero class in the H_2(W)^L-graded component of the equivariant skein lasagna module S(W;L)/torsion in bidegree (α, α·α). In particular, ascertain for which smooth, compact, oriented 4-manifolds W, framed oriented links L ⊂ ∂W, and classes α ∈ H_2(W)^L the resulting bounds are sharp or nontrivial.

Background

Corollary B introduces bounds on the Euler characteristic of surfaces in a 4‑manifold W in terms of an invariant q^N_min(W,L,α) extracted from the equivariant skein lasagna module S(W;L). Theorem A ensures nonvanishing modulo torsion of the associated surface classes under a homological diversity condition, which guarantees that q^N_min is defined (possibly taking the value −∞).

The authors note that, beyond certain classes (e.g., positive links in B^4), the practical strength of these bounds is unclear. Establishing when these bounds are sharp or informative requires computing or estimating q^N_min(W,L,α) for broader families of 4‑manifolds and links and comparing with known surfaces realizing minimal genus.