Tri-plane diagrams for simple surfaces in $S^4$

TL;DR

This paper determines minimal crossing numbers for certain unknotted nonorientable surfaces in 4-sphere and converts existing diagrams to tri-plane diagrams, providing new insights into their complexity.

Contribution

It establishes the minimal crossing numbers for nonorientable unknotted surfaces in $S^4$ and translates Yoshikawa's knotted surface diagrams into tri-plane diagrams, enhancing understanding of their structure.

Findings

Minimal crossing number for $ ext{P}^{n,m}$ is $ ext{max}igrace{1,|n-m|}$.

Converted Yoshikawa's diagrams to tri-plane diagrams.

Provided minimal bridge numbers and crossing number bounds for surfaces in the table.

Abstract

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.