Publication: On the classification of 3-bridge links.
Loading...
Full text at PDC
Publication Date
2012
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Soc. Colombiana Mat.
Citations
Exportar
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?
Description
UCM subjects
Unesco subjects
Keywords
Citation
G. Burde and H. Zieschang, Knots, Walter de Gruyter, New York, USA,1985.
A. Cavicchioli and B. Ruini, Special Representations for n{Bridge Links, Discrete Comput. Geom. 12 (1994), no. 1, 9-27.
P. Cromwell, Knots and Links, Cambridge University Press, Cambridge,United Kingdom, 2004.
D. M. Tejada H. M. Hilden, J. M. Montesinos and M. M. Toro, Mariposas y 3{variedades, Revista de la Academia Colombiana de Ciencias Exactas, Fsicas y Naturales 28 (2004), 71-78 (sp).
D. M. Tejada H. M. Hilden, J. M. Montesinos and M. M. Toro, Representing 3{Manifolds by Triangulations of S 3, Revista Colombiana de Matematicas 39 (2005), no. 2, 63-86.
D. M. Tejada H. M. Hilden, J. M. Montesinos and M. M. Toro, Fox Coloured Knots and Triangulations of S 3 , Math Proc. Camb. Phil. Soc. 141 (2006), no. 3, 443-463.
H. M. Hilden, J. M. Montesinos, D. M. Tejada, and M. M. Toro, A new Representation of Links: Butterflies, arXiv:1203.2045v1.
M. H. Hilden, J. M. Montesinos, D. M. Tejada, and M. M. Toro, Knots Butterflies and 3{Manifolds, Publicado en: Disertaciones del Seminario de Matematicas Fundamentales de la UNED (Universidad de Educacion a Distancia) (Madrid, España), vol. 33, 2005, pp. 1-22.
K. Kawauchi, A Survey of Knot Theory, Birkh�auser Verlag, Basel, Switzerland, 1996.
J. M. Montesinos, Sobre la representacion de variedades tridimensionales, Tech. report, Universidad Complutense de Madrid, 1978.
J. M. Montesinos, Calidoscopios y 3{variedades, Tech. report, Universidad Nacional de Colombia, Medelln, Colombia, 2003.
O. Morikawa, A Class of 3{Bridge Knots I, Math. Semin. Notes, Kobe Univ. 9 (1981), 349-369.
O. Morikawa, A Class of 3{Bridge Knots II, Yojohama Mathematical Journal 30 (1982), 53-72.
K. Murasugi, Knot Theory and its Applications, Birkh�auser, Boston. USA, 1996.
S. Negami, The Minimum Crossing of 3{Bridge Links, Osaka J. Math. 21 (1984), no. 3, 477-487.
S. Negami and K. Okita, The Splittability and Triviality of 3{Bridge Links, Trans. Amer. Math. Soc. 289 (1985), no. 1, 253-280.