Minimum curve-complex distance condition for strengthened lower bounds on splitting complexities

Determine whether, for the vertex pairs that realize the splitting distances used in this paper—namely, (i) the Hatcher–Thurston cut-system distance D^{HT}(Σ_g) for Heegaard splittings of closed 3‑manifolds and (ii) the dual curve distance D(Σ_{c,s_1,s_2}) for bridge splittings of handlebody‑knots—the minimum curve complex distance min{ d(α, β) : α is a curve in one vertex, β is a curve in the other } is always strictly greater than 1. Establishing this would guarantee that the stronger lower bounds from Section 4 apply uniformly to these splitting distances and the associated complexities.

Background

The paper derives general lower bounds on distances in admissible multi‑curve complexes in terms of the number of components N and the minimum pairwise distance k in the curve complex between components of two vertices (Theorem A). These bounds are stronger when k≥2.

In the applications to closed 3‑manifolds (via the Hatcher–Thurston cut system) and to handlebody‑knots (via the dual curve complex), the authors use these bounds to estimate splitting distances and define associated complexities. They note that the strengthened bounds are useful provided the minimum curve‑complex distance between the defining vertex sets exceeds 1, but they are unsure if that condition always holds in these settings.

If the minimum distance is 0, the bounds from Section 4 give no information, prompting alternative estimates independent of k. Clarifying whether the minimum distance is always greater than 1 would settle when the stronger bounds can be universally applied.