We adapt work of Kirby-Thompson and Zupan to define an integer invariant $\mathcal{ L}(\mathcal{ T})$ of a bridge trisection $\mathcal{ T}$ of a smooth surface $\mathcal{ K}$ in $S^4$ or $B^4$. We show that when $\mathcal{ L}(\mathcal{ T})=0$, 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 $\mathcal{ L}(\mathcal{ T})$. Consequently, $\mathcal{ L}$ can be arbitrarily large.
Categories: math.GT