Tri-plane diagrams for simple surfaces in S⁴

2023/02/16 by Wolfgang Allred, Allred, Wolfgang, Manuel Gutiérrez Aragón +18 · 1 citation
Computer Science · Mathematics · #57K10 #Advanced Combinatorial Mathematics #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2302.08431

openalex publication_date 2023/02/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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