Tri-plane diagrams for simple surfaces in S4

Abstract

Meier and Zupan proved that an orientable surface [Formula: see text] in [Formula: see text] admits a tri-plane diagram with zero crossings if and only if [Formula: see text] is unknotted, so that the crossing number of [Formula: see text] is zero. We determine the minimal crossing numbers of nonorientable unknotted surfaces in [Formula: see text], proving that [Formula: see text], where [Formula: see text] denotes the connected sum of [Formula: see text] unknotted projective planes with normal Euler number [Formula: see text] and [Formula: see text] unknotted projective planes with normal Euler number [Formula: see text]. 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.