Pants distances of knotted surfaces in 4-manifolds

TL;DR

This paper introduces a new invariant called pants distance for knotted surfaces in 4-manifolds, generalizing existing complexity measures and providing insights into the topology of these pairs.

Contribution

It defines a pants distance invariant for knotted surfaces in 4-manifolds and explores its implications, including bounds related to multisection genus and exact calculations for specific examples.

Findings

If the pants distance is below a certain bound, the topology is simple.

Exact invariants are computed for examples like spun lens spaces.

Characterization of genus two quadrisections with low distance.

Abstract

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 (4-manifold, surface) pair has a 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.