Finiteness of q^N_min for general 4-manifolds

Ascertain whether the invariant q^N_min(W,L,α), defined as 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), is finite for broad classes of smooth, compact, oriented 4-manifolds W and framed oriented links L ⊂ ∂W, or whether q^N_min(W,L,α) equals −∞ for “most” 4‑manifolds.

Background

The function q^N_min(W,L,α) controls the upper bound on χ(S) in Corollary B via χ(S) ≤ (−N·([S]·[S]) − q^N_min(W,L,[S]))/(N−1). If q^N_min(W,L,α)=−∞, the resulting bound is vacuous and suggests, as the authors remark, a potential connection with vanishing phenomena for undeformed skein lasagna modules.

The authors emphasize that present evidence does not exclude the possibility that q^N_min is −∞ in many cases; clarifying finiteness versus −∞ would determine whether these invariants yield substantive genus bounds across large classes of 4‑manifolds.