Tri-plane diagrams for simple surfaces in $S^4$
| Autor: | Allred, Wolfgang, Aragón, Manuel, Dooley, Zack, Goldman, Alexander, Lei, Yucong, Martinez, Isaiah, Meyer, Nicholas, Peters, Devon, Warrander, Scott, Wright, Ana, Zupan, Alexander |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 25 pages, 29 figures |
| Databáze: | arXiv |
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