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.Comment: 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.htm