Kirby–Thompson distance for trisections of knotted surfaces

Abstract

We adapt work of Kirby–Thompson and Zupan to define an integer invariant (Formula presented.) of a bridge trisection (Formula presented.) of a smooth surface (Formula presented.) in (Formula presented.) or (Formula presented.). We show that when (Formula presented.), then the surface (Formula presented.) is unknotted. We also show that for a trisection (Formula presented.) of an irreducible surface, bridge number produces a lower bound for (Formula presented.). Consequently (Formula presented.) can be arbitrarily large.