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On the classification of 3-bridge links.

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Hilden, Hugh Michael and Montesinos Amilibia, José María and Tejada Jiménez, Débora María and Toro Villegas, Margarita María (2012) On the classification of 3-bridge links. Revista Colombiana de Matemáticas, 46 (2). pp. 113-144. ISSN 0034-7426

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Abstract

Using a new way to represent links, that we call a butter
y 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 innite number of 3-bridge diagrams, so we dene 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-butter y of L, and call it the butter y presentation of L. This presentation extends, in a natural way, the well known Schubert classication of 2-bridge links.
We obtain necessary and sucient conditions for a triple (p=n; q=m; s=l) to correspond to a 3-butter y 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-butter
y 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 0 =n0 ; q0 =m0 ; s0 =l0) represent the same 3-bridge link?


Item Type:Article
Uncontrolled Keywords:Links; 3-bridge links; Bridge presentation; Link diagram; 3-butterfly; Butterfly presentation
Subjects:Sciences > Mathematics > Algebraic geometry
Sciences > Mathematics > Combinatorial analysis
Sciences > Mathematics > Topology
ID Code:21531
Deposited On:24 May 2013 15:45
Last Modified:12 Dec 2018 15:12

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