Tri-plane diagrams for simple surfaces in \(S^4\)

Meier and Zupan proved that an orientable surface \(\mathcal{K}\) in \(S^4\) admits a tri-plane diagram with zero crossings if and only if \(\mathcal{K}\) is unknotted, so that the crossing number of \(\mathcal{K}\) is zero. We determine the minimal crossing numbers of nonorientable unknotted surfaces in \(S^4\), proving that \(c(\mathcal{P}^{n,m}) = \max\{1,|n-m|\}\), where \(\mathcal{P}^{n,m}\) denotes the connected sum of \(n\) unknotted projective planes with normal Euler number \(+2\) and \(m\) unknotted projective planes with normal Euler number \(-2\). In addition, we convert Yoshikawa's table of knotted surface ch-diagrams to tri-plane diagrams, finding the minimal bridge number for each surface in the table and providing upper bounds for the crossing numbers.