Kirby-Thompson distance for trisections of knotted surfaces.

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