Mathematics, Department of


Date of this Version



Published (2023) Journal of Knot Theory and its Ramifications, 32 (6), art. no. 2350041-1, .


Used by permission


Meier and Zupan proved that an orientable surface K in S4 admits a tri-plane diagram with zero crossings if and only if K is unknotted, so that the crossing number of K is zero. We determine the minimal crossing numbers of nonorientable unknotted surfaces in S4, proving that c(Pn,m) = max{1, |nm|}, where Pn,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.