GEOMETRIC KNOT SPACES AND POLYGONAL ISOTOPY
GEOMETRIC KNOT SPACES AND POLYGONAL ISOTOPY
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几何结空间和多边形同位素
DOI：
10.1142/s0218216501000834
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发表时间：
1999
影响因子：
0.5
通讯作者：
J. A. Calvo
中科院分区：
文献类型：
--
作者：
J. A. Calvo
嵌入在三维空间中的n边多边形的空间由一个光滑流形组成，其中的点对应于分段线性或“几何”结，而路径对应于保持这些结的几何结构的等距。给出了n=6和n=7时这些空间的拓扑图。在这两种情况下，每个节点空间由五个分量组成，但只包含三种(当n=6时)或四种(当n=7时)拓扑节点类型。因此，“几何纽结等价”严格强于拓扑等价。这一点由六角形树叶和七角形八结证明，这与它们的拓扑对应不同，是不可逆的。将这些结果推广到≥8的情况也将被讨论。
The space of n-sided polygons embedded in three-space consists of a smooth manifold in which points correspond to piecewise linear or "geometric" knots, while paths correspond to isotopies which preserve the geometric structure of these knots. The topology of these spaces for the case n=6 and n=7 is described. In both of these cases, each knot space consists of five components,but contains only three (when n=6) or four (when n=7) topological knot types. Therefore "geometric knot equivalence" is strictly stronger than topological equivalence. This point is demonstrated by the hexagonal trefoils and heptagonal figure-eight knots, which, unlike their topological counterparts, are not reversible. Extending these results to the cases n≥8 will also be discussed.