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 and lt;n≤ p, 0 and lt;m≤ q and 0 and lt;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?

• Autores: Hilden, Hugh Michael; Montesinos, José María; Tejada, Débora María; Toro, Margarita María
• Tipo de recurso: Article of journal
• Fecha de publicación: 2012
• Institución: Universidad Nacional de Colombia
• Repositorio: Universidad Nacional de Colombia
• Idioma: spa
• OAI Identifier: oai:repositorio.unal.edu.co:unal/49375
• Acceso en línea: https://repositorio.unal.edu.co/handle/unal/49375; http://bdigital.unal.edu.co/42832/
• Palabra clave: Enlaces; enlaces de 3 puentes; presentación en puentes; diagrama deenlace; 3-mariposa; presentación en mariposa; 57M25; 57M27; Links; 3-bridge links; Bridge presentation; Link diagram; 3-butterfly; Butterfly presentation
• Rights: openAccess
• License: Atribución-NoComercial 4.0 Internacional