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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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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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.</description><identifier>EISSN: 2331-8422</identifier><language>eng</language><publisher>Ithaca: Cornell University Library, arXiv.org</publisher><subject>Knot theory ; Upper bounds</subject><ispartof>arXiv.org, 2023-02</ispartof><rights>2023. This work is published under http://arxiv.org/licenses/nonexclusive-distrib/1.0/ (the “License”). 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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 \(-2\). 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subjects | Knot theory Upper bounds |
title | Tri-plane diagrams for simple surfaces in \(S^4\) |
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