1999/04/08 by Jorge Alberto Calvo, Calvo, Jorge Alberto · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · #57M25 #Connective tissue disorders research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.math/9904037
openalex publication_date 1999/04/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 is also discussed.