Geometric Knot Spaces and Polygonal Isotopy
Abstract
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.
Keywords
Cite
@article{arxiv.math/9904037,
title = {Geometric Knot Spaces and Polygonal Isotopy},
author = {Jorge Alberto Calvo},
journal= {arXiv preprint arXiv:math/9904037},
year = {2009}
}
Comments
AMS LaTeX, 23 pages, 14 figures, 1 table; submitted to Journal of Knot Theory and its Ramifications, and to Proceedings of the International Knot Theory Meeting (Knots in Hellas 1998), Delphi, Greece, 7 - 15 August 1998. Also available from http://www.williams.edu/Mathematics/jcalvo/abstract.html