Montesinos Amilibia, José María
(1976)
*Minimal plat representations of prime knots and links are not unique.*
Canadian Journal of Mathematics-Journal Canadien de Mathématiques, 28
(1).
pp. 161-167.
ISSN 0008-414X

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Official URL: http://cms.math.ca/cjm/v28/cjm1976v28.0161-0167.pdf

## Abstract

J. S. Birman [same J. 28 (1976), no. 2, 264–290] has shown that any two plat representations of a link in S3 are stably equivalent and that stabilization is a necessary feature of the equivalence for certain composite knots. She has asked whether all 2n-plat representations of a prime link are equivalent. The author provides a negative answer, by exhibiting an infinite collection of prime knots and links in S3 in which each element L has at least two minimal and inequivalent 6-plat representations. In addition, as an application of another result of Birman [Knots, groups and 3-manifolds (Papers dedicated to the memory of R. H. Fox), pp. 137–164, Ann. of Math. Studies, No. 84, Princeton Univ. Press, Princeton, N.J., 1975], the 2-fold cyclic covering spaces of S3 branched over such links L form further examples of closed, orientable, prime 3-manifolds having inequivalent minimal Heegaard splittings, which were first constructed by Birman, F. González-Acuña and the author [Michigan Math. J. 23 (1976), no. 2, 97–103].

Item Type: | Article |
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Uncontrolled Keywords: | Topology of general 3-manifolds |

Subjects: | Sciences > Mathematics > Differential geometry |

ID Code: | 17266 |

References: | J. S. Birman and H. M. Hilden, On the mapping class group of closed, orientable surfaces as covering spaces, Annals of Math. Studies 66, 81-115. J. S. Birman, On the equivalence of Heegaard splittings of closed, orientable 3-manifolds, Knots, Groups and 3-Manifolds (L. Neuwirth, Editor), Annals of Math. Studies 84 (1975), 137-164. J. S. Birman, F. Gonzalez-Acufïa and J. M. Montesinos, Heegaard splittings of prime 3-manifolds are not unique, to appear, Michigan Math. J. J. S. Birman, Braids, links and mapping class groups, Annals of Math. Studies 82 (1975). J. S. Birman, On the stable equivalence of plat representations of knots and links, to appear, Can. J. Math. R. Engmann, Nicht-hom'ôomorphe Heegaard-Zerlegungen vom Geschlecht 2 der zusammenhdngendem Summe zweier Linsenrâume, Abh. Math. Sem. Univ. Hamburg 35(1970), 33-38. J. M. Montesinos, Sobre la conjetura de Poincaré y los recubridores ramificados sobre un nudo, Tesis doctoral, Madrid, 1971. J. M. Montesinos, Variedades de Seifert que son recubridores ciclicos ramificados de dos hojas, Boletin Soc. Mat. Mexicana 18 (1973), 1-32. K. Reidemeister, Zur dreidimensionalen Topologie, Abh. Math. Sem. Univ. Hamburg 9 (1933), 189-194. H. Seifert, Topologie dreidimensionaler gefaserter Raume, Acta Math. 60 (1933), 147-238. J. Singer, Three dimensional manifolds and their Heegaard diagrams, Trans. Amer. Math. Soc. 85 (1933), 88-111. 0. Ja. Viro, Linkings, 2-sheeted branched coverings, and braids, Math. U.S.S.R. Sbornik 16 (1972), 222-236 (English translation). F. Waldhausen, Eine Klasse von 3-dimensionalen Mannigfaltigkeiten II, Invent. Math 4 (1967), 87-117. F. Waldhausen, Uber Involutionen der 3-Sphare, Topology 8 (1969), 81-91. |

Deposited On: | 29 Nov 2012 09:54 |

Last Modified: | 07 Feb 2014 09:44 |

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