Geometric Knot Spaces and Polygonal Isotopy

Details Geometric Knot Spaces and Polygonal Isotopy Geometric Knot Spaces and Polygonal Isotopy

1999-04-08

arxiv.org/abs/math/9904037v2

Mathematics

Geometric Topology

AMS LaTeX, 23 pages, 14 figures, 1 table; submitted to Journal of Knot Theory and its Ramifications, and to Proceedings of the

Scientific paper

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 \ge 8 is also discussed.

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