TRI-PLANE DIAGRAMS FOR SIMPLE SURFACES IN S⁴

Authors

, University of Nebraska-LincolnFollow
MANUEL ARAGÓN, Universidad de los Andes
ZACK DOOLEY, Reed College
ALEXANDER GOLDMAN, Skidmore College
YUCONG LEI, University of Michigan-Ann Arbor
ISAIAH MARTINEZ, California State University, Fresno
NICHOLAS MEYER, University of Nebraska-LincolnFollow
DEVON PETERS, Oakland University
SCOTT WARRANDER, University of Edinburgh
Ana Wright, University of Nebraska-LincolnFollow
Alex Zupan, University of Nebraska-LincolnFollow

Document Type

Article

Date of this Version

2-16-2023

Citation

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

Comments

Used by permission

Abstract

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(P^n,m) = max{1, |n−m|}, where 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.