The structure of a minimal \(n\)-chart with two crossings. I: Complementary domains of \({\Gamma}_1 \cup {\Gamma}_{n - 1}\). (English) Zbl 1408.57025
Take a closed surface \(F \subset \mathbb R^4\), i.e. 2-knot. Let \(p_1:\mathbb R^4 \to \mathbb R^3\) be a generic projection. \(p_1|_F\) has singularities which are double points, triple points and branch points. As the next step consider a plane \(P \subset \mathbb R^3\) such that \(p_1(F) \subset \mathbb R^3 \setminus P\) and a projection \(p_2:\mathbb R^3 \to P\). Then the image of \(p_2 p_1\) of the fold lines and cusps forms an oriented graph \(\Gamma\) called the chart. Every vertex of \(\Gamma\) has degree 1 (black vertex), 4 (crossing) or 6 (white vertex). A chart of degree \(n\) has edges labeled from 1 to \(n-1\). From a chart we can construct the closed surface.
The paper, at its introduction, has a list of terms and symbols that is quite useful. Therefore we will not introduce other terms and notions but quote a sentence from the paper: “We investigate the structure of minimal charts with two crossings and give an enumeration of the charts with two crossings.”
The paper, at its introduction, has a list of terms and symbols that is quite useful. Therefore we will not introduce other terms and notions but quote a sentence from the paper: “We investigate the structure of minimal charts with two crossings and give an enumeration of the charts with two crossings.”
Reviewer: Ivan Ivanšić (Zagreb)
MSC:
| 57Q45 | Knots and links in high dimensions (PL-topology) (MSC2010) |
| 57Q35 | Embeddings and immersions in PL-topology |
References:
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