Keywords
space polygons
knot spaces
knot invariants
nudos poligonales
polígonos espaciales
espacios de nudos
invariantes de nudos
How to Cite
Abstract
The space of n-sided polygons embedded in euclidean 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.
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