### Impacto

Montesinos Amilibia, José María and Whitten, Wilbur Carrington
(1986)
*Constructions of two-fold branched covering spaces.*
Pacific Journal of Mathematics, 125
(2).
pp. 415-446.
ISSN 0030-8730

PDF
Restringido a Repository staff only hasta 31 December 2020. 2MB |

Official URL: http://msp.org/pjm/1986/125-2/pjm-v125-n2-p11-s.pdf

## Abstract

By equivariantly pasting together exteriors of links in S3 that are invariant under several different involutions of S3, we construct closed orientable 3-manifolds that are two-fold branched covering spaces of S3 in distinct ways, that is, with different branch sets. Sufficient conditions are given to guarantee when the constructed manifold M admits an induced involution, h, and when M∕h≅S3. Using the theory of characteristic submanifolds for Haken manifolds with incompressible boundary components, we also prove that doubles, D(K,ρ), of prime knots that are not strongly invertible are characterized by their two-fold branched covering spaces, when ρ≠0. If, however, K is strongly invertible, then the manifold branch covers distinct knots. Finally, the authors characterize the type of a prime knot by the double covers of the doubled knots, D(K;ρ,η) and D(K∗;ρ,η), of K and its mirror image K∗ when ρ and η are fixed, with ρ≠0 and η ∈{−2,2}.

Item Type: | Article |
---|---|

Uncontrolled Keywords: | strongly invertible knot; symmetric links; exteriors of links; involutions of S 3 ; two-fold branched covering spaces of S 3 ; surgery on a trefoil knot; characteristic submanifolds; Haken manifolds; prime knots; doubled knots |

Subjects: | Sciences > Mathematics > Topology |

ID Code: | 17182 |

References: | J. Birman, F. Gonzalez-Acuna and J. M. Montesinos, Heegaard splittings of prime 3-manifolds are not unique, Michigan Math. J., 23 (1976), 97-103. Steven A. Bleiler, Knots prime on many strings, Trans. Amer. Math. Soc, (to appear). M. Boileau, F. Gonzalez-Acuha and J. M. Montesinos, Surgery on double knots and symmetries, (to appear). R. H. Bing and J. M. Martin, Cubes with knotted holes, Trans. Amer. Math. Soc, 155 (1971), 217-231. G. Burde and K. Murasugi, Links and Seifert fiber spaces, Duke Math. J., 37 (1970), 89-93. S. Furusawa and M. Sakuma, Dehn surgery on symmetric knots, Math. Sem. Notes, 11 (1983), 179-198. F. Gonzalez Acuña, Ph.D. Thesis, Princeton University, 1969. C. McA. Gordon and Wolfgang Heil, Cyclic normal subgroups of fundamental groups of 3-manifolds,Topology,14 (1975), 305-309. C. McA. Gordon and R. A. Litherland, Incompressible surfaces in branched coverings, Proceedings of the Symposium on the Smith Conjecture, Columbia University, (1978), 139-152. R. Hartley, Knots and involutions, Math. Zeit., 171 (1980),175-185. C. D. Hodgson, Involutions and isotopies of lens spaces, MS Thesis, University of Melbourne,1981. W. Jaco, Lectures on Three-Manifold Topology, Regional Conference Series43, Amer. Math. Soc,Providence, R. L,1980. W. Jaco and P. Shalen, Seifert Fibered Spaces in 3-Manifolds, Memoirs of the Amer. Math. Soc,Vol. 21,No. 220,Amer. Math. Soc,Providence, R. I.,1979. P. K. Kim, Involutions on Klein spaces M(p, q), Trans. Amer. Math. Soc,268 (1981), 377-409. Kim and J. L. Tollefson, PL involutions of fibered 3-manifolds, Trans. Amer. Math. Soc,232 (1977),221-237. Kim and J. L. Tollefson, Splitting the PL involutions of non-prime 3-manifolds, Michigan Math. J., 27 (1980),259-274. K. W. Kwun and J. L. Tollefson, Extending PL involutions of a compact manifold, Amer. J. Math., 99 (1977),995-1001. Jose M. Montesinos, Variedades de Seifert que son recubridores ciclicos ramificados de dos hojas, Bol. Soc. Math. Mexicana (2), 18 (1973), 1-32. Jose M. Montesinos, Sugery on links and double branched covers, Knots, Groups, and 3-Manifolds, Ann. of Math. Studies, 84, 227-259, Princeton University Press, Princeton, N.J.,1975. Jose M. Montesinos, Sobre la representacion de variedades tridimensionales, unpublished preprint (1975). W. H. Meeks and P. Scott, Finite group actions on 3-manifolds, preprint. Jose M. Montesinos and W. Whitten, Constructions of two-fold branchedcovering spaces, Abstracts Amer. Math. Soc, 4 (Mar., 1983), Abstract 802-57-11, p. 178. J. H. Rubinstein, Representations of some 3-manifolds as 2-fold cyclicbranched covers ofS3, Notices Amer. Math. Soc,25 (1978), Abstract 78T-G7, A-18. M. Sakuma, Surface bundles over S1 which are 2-fold branched cyclic coverings ofS3, Math. Sem.notes, 9 (1981), 159-180. H. Schubert,Knotenund Vollringe, Acta Math., 90 (1953), 131-286. M. Takahashi, Two knots with the same two fold branched covering space, Yokohama Math. J., 25 (1977), 91-99. M. Takahashi, On homology spheres obtained by surgery on the figure-eight knot, Proc Sympos. Res. Inst. Math. Sci. Univ. Kyoto, 309 (1977) (in Japanese). W. Thurston, The Geometry and Topology of 3-Manifolds, Lecture notes, Princeton University, 1978-1979. Tollefson, involutions on S X S2 and other 3-manifolds, Trans. Amer. Math. Soc, 183 (1973), 139-152. Tollefson, Involutions of Seifert fiber spaces, Pacific J. Math., 74 (1978), 519-529. Tollefson, Involutions of sufficiently large 3-manifolds, Topoloy, 20 (1981), 323-352. F. Waldhausen, Eine Klasse von 3-dimensionalen Mannigfaltigkeiten I, II, Invent. Math., 3 (1967), 308-333; and Invent. Math., 4 (1967), 87-117. F. Waldhausen, Uber Involutionen der 3-Sphre, Topology,8 (1969), 81-91. W. Whitten, Algebraic and geometric characterizationsof knots, Invent. Math., 26 (1974), 259-270. W. Whitten, Inverting doubleknots, Pacific J. Math,97 (1981), 209-216. H. Zieschang, On extensions of fundamental groups of surfaces and related groups, Bull. Amer. Math. Soc,77 (1971), 1116-1119. H. Zieschang, Addendum to: On extensions of fundamental groups of surfaces and related groups, Bull. Amer. Math. Soc,80 (1974), 366-367. |

Deposited On: | 23 Nov 2012 11:52 |

Last Modified: | 07 Feb 2014 09:43 |

Repository Staff Only: item control page