Computability of the cut-complex distance for a given trisection

Determine whether the distance in the cut complex associated with a given trisection of a closed orientable connected smooth 4-manifold is computable.

Background

The paper proves that the Kirby–Thompson L-invariant of a closed orientable connected smooth 4-manifold is uncomputable when the input is a PL triangulation. The authors explain that this result resolves the second part of Problem 4.116 in the K3 problem list, while the first part remains unresolved. That first part concerns the computability of a certain distance in the cut complex associated with an individual trisection, in contrast to the Kirby–Thompson L-invariant, which is obtained by minimizing this distance over all trisections of the manifold.