Teoría de nudos geométricos e isotopía poligonal

• Autores: Jorge Alberto Calvo Soto
• Localización: Revista de Matemática: Teoría y Aplicaciones, ISSN 2215-3373, ISSN-e 2215-3373, Vol. 8, Nº. 2, 2001, págs. 101-130
• Idioma: español
• DOI: 10.15517/rmta.v8i2.204
• Enlaces
  • Texto completo (pdf)
• Resumen
  • español

    El espacio de los pol´?gonos de n lados, inmersos en el espacio eucl´?deo de tres dimensiones,consiste de una variedad suave en la cual los puntos corresponden a nudos lineales a trozoso “geom´etricos”, mientras que los arcos corresponden a isotop´?as que preservan la estructurageom´etrica de esos nudos. Se describe la topolog´?a de estos espacios para los casos n = 6y n = 7. En ambos casos, cada espacio consta de cinco componentes, aunque contiene s´olotres (cuando n = 6) o cuatro (cuando n = 7) tipos topol´ogicos de nudos. Por lo tanto la“equivalencia geom´etrica de nudos” es estrictamente m´as fuerte que la equivalencia topol´ogica.Este hecho se demuestra con el nudo tr´ebol hexagonal y el nudo doble heptagonal, los cuales,a diferencia de sus contrapartes topol´ogicas, no son reversibles. Se discutir´an tambi´en lasextensiones de estos resultados a los casos n 8.Palabras clave:  nudos poligonales, pol´?gonos espaciales, espacios de nudos, invariantesde nudos.

  • English

    The space of n-sided polygons embedded in euclidean three-space consists of a smoothmanifold in which points correspond to piecewise linear or “geometric” knots, while pathscorrespond to isotopies which preserve the geometric structure of these knots. The topologyof these spaces for the case n = 6 and n = 7 is described. In both of these cases, each knotspace 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 thantopological equivalence. This point is demonstrated by the hexagonal trefoils and heptagonalfigure-eight knots, which, unlike their topological counterparts, are not reversible. Extendingthese results to the cases n 8 will also be discussed.Keywords:  polygonal knots, space polygons, knot spaces, knot invariants.

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