On the classification of 3-bridge links

Using a new way to represent links, that we call a butterfly representation, we assign to each 3-bridge link diagram a sequence of six integers, collected as a triple (p/n, q/m, s/l), such that p ≥ q ≥ ≥ s ≥ 2, 0 < n ≤ p, 0 < m ≤ q and 0 < l ≤ s. For each 3-bridge link there exists an infinite number of 3-bridge diagrams, so we define an order in the set (p/n, q/m, s/l) and assign to each 3-bridge link L the minimum among all the triples that correspond to a 3-butterfly of L, and call it the butterfly presentation of L. This presentation extends, in a natural way, the well-known Schubert classification of 2-bridge links. We obtain necessary and sufficient conditions for a triple (p/n, q/m, s/l) to correspond to a 3-butterfly and so, to a 3-bridge link diagram. Given a triple (p/n, q/m, s/l) we give an algorithm to draw a canonical 3-bridge diagram of the associated link. We present formulas for a 3-butterfly of the mirror image of a link, for the connected sum of two rational knots and for some important families of 3-bridge links. We present the open question: When do the triples (p/n, q/m, s/l) and (p’/n’, q’/m’, s’/l’) represent the same 3-bridge link?

• Fecha de publicación: 2012
• Institución: Ministerio de Ciencia, Tecnología e Innovación
• Repositorio: Repositorio Minciencias
• Idioma: eng
• OAI Identifier: oai:repositorio.minciencias.gov.co:20.500.14143/22006
• Acceso en línea: https://repositorio.minciencias.gov.co/handle/20.500.14143/22006
• Palabra clave: Teoría de los números; Bridge links; Bridge presentation; Link diagram; Butterfly; Butterfly presentation; Variables reales; Topología algebraica; Homomorfismos; Modelos matemáticos; Diagramas de curvas
• License: http://purl.org/coar/access_right/c_f1cf