Making Curves Minimally Crossing by Reidemeister Moves
Making Curves Minimally Crossing by Reidemeister Moves
复制标题
通过 Reidemeister 移动使曲线最小化交叉
DOI：
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发表时间：
1997
期刊：
影响因子：
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通讯作者：
A. Schrijver
中科院分区：
文献类型：
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作者：
M. D. Graaf;A. Schrijver
令C1，？，是可三角化曲面上的闭曲线系。该系统被称为minimally crossing，如果每个curveCi有一个最小数量的self-intersects之间的所有curveC？如果每个曲线对Ci，Cj在所有曲线对C中有最小的交点数，则C是自由同伦的。i，C？j自由同伦于Ci，Cj（i，j=1，？，k，i？j）。这个系统称为正则的，如果这些曲线至少经过两次的每个点正好经过两次，并形成一个交叉。我们表明，我们可以使任何定期系统最小交叉应用Reidemeister移动这样一种方式，在每次移动的交叉点的数量不会增加。它意味着一个有限的算法，使一个给定的曲线系统最小交叉的Reidemeister移动。
LetC1, ?, Ckbe a system of closed curves on a triangulizable surfaceS. The system is calledminimally crossingif each curveCihas a minimal number of self-intersections among all curvesC?ifreely homotopic toCiand if each pairCi,Cjhas a minimal number of intersections among all curve pairsC?i, C?jfreely homotopic toCi, Cjrespectively (i, j=1, ?, k, i?j). The system is called regular if each point traversed at least twice by these curves is traversed exactly twice, and forms a crossing. We show that we can make any regular system minimally crossing by applying Reidemeister moves in such a way that at each move the number of crossings does not increase. It implies a finite algorithm to make a given system of curves minimally crossing by Reidemeister moves.