Pants distances of knotted surfaces in 4-manifolds

We define a pants distance for knotted surfaces in 4-manifolds, which generalizes the complexity studied by Blair-Campisi-Taylor-Tomova for surfaces in the 4-sphere.

We determine that if the distance computed on a given diagram does not surpass a theoretical bound in terms of the multisection genus, then the pair (X, F) admits a standard form (i.e., has simple topology).

Furthermore, we calculate the exact values of our invariants for many new examples, such as the spun lens spaces.

We provide a characterization of genus two quadrisections with distance at most six.