Tri-plane diagrams for simple surfaces in \(S^4\)

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 surfac...

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Veröffentlicht in:arXiv.org 2023-02
Hauptverfasser: Allred, Wolfgang, Aragón, Manuel, Dooley, Zack, Goldman, Alexander, Yucong Lei, Martinez, Isaiah, Meyer, Nicholas, Peters, Devon, Warrander, Scott, Wright, Ana, Zupan, Alexander
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creator Allred, Wolfgang
Aragón, Manuel
Dooley, Zack
Goldman, Alexander
Yucong Lei
Martinez, Isaiah
Meyer, Nicholas
Peters, Devon
Warrander, Scott
Wright, Ana
Zupan, Alexander
description 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.
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Upper bounds
title Tri-plane diagrams for simple surfaces in \(S^4\)
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