Minimum curve-complex distance condition for strengthened lower bounds

Determine whether, in the settings considered for defining splitting distances and complexities for 3‑manifolds and handlebody‑knots via admissible multi‑curve complexes (including the dual curve complex, pants complex, and Hatcher–Thurston cut system), the minimum curve complex distance between any component of a vertex defining one handlebody and any component of a vertex defining the other handlebody is always greater than 1. Establishing this would confirm when the general lower bound d ≥ k (N − 1) + 1 yields strictly stronger estimates for splitting distance and the associated complexities.

Background

The paper develops general lower bounds relating distances in admissible multi‑curve complexes associated to a splitting surface to distances in the curve complex. Specifically, Theorem A shows that if every component of one vertex is at least distance k in the curve complex from every component of another vertex, then the complex distance is at least k(N − 1) + 1, where N is the number of components in each vertex.

In applications to defining complexities for Heegaard splittings of 3‑manifolds and bridge positions of handlebody‑knots, these bounds provide immediate lower bounds on splitting distances. The authors note that the bounds are stronger when the minimum curve complex distance k between components of the defining vertices exceeds 1, but it is not established whether this condition universally holds in these contexts. If k = 0, the bound provides no information, motivating alternative estimates independent of k.