Kirby–Thompson distance for trisections of knotted surfaces

We adapt work of Kirby–Thompson and Zupan to define an integer invariant L(T)$\mathcal {L}(\mathcal {T})$ of a bridge trisection T$\mathcal {T}$ of a smooth surface S$S$ in S4$S^4$ or B4$B^4$. We show that when L(T)=0$\mathcal {L}(\mathcal {T})=0$, then the surface S$S$ is unknotted. We also show that for a trisection T$\mathcal {T}$ of an irreducible surface, bridge number produces a lower bound for L(T)$\mathcal {L}(\mathcal {T})$. Consequently L$\mathcal {L}$ can be arbitrarily large.