Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-25T16:48:11.963Z Has data issue: false hasContentIssue false
  • English
  • Français

Dehn surgeries on some classical links

Part of: Low-dimensional topology Differential topology Special aspects of infinite or finite groups

Published online by Cambridge University Press:  19 January 2011

Alberto Cavicchioli ,
Fulvia Spaggiari  and
Agnese Ilaria Telloni
Alberto Cavicchioli
Affiliation:
Dipartimento di Matematica, Università di Modena e Reggio Emilia, Via Campi 213/B, 41100 Modena, Italy (cavicchioli.alberto@unimore.it; spaggiari.fulvia@unimore.it; agneseilaria.telloni@unimore.it)
Fulvia Spaggiari
Affiliation:
Dipartimento di Matematica, Università di Modena e Reggio Emilia, Via Campi 213/B, 41100 Modena, Italy (cavicchioli.alberto@unimore.it; spaggiari.fulvia@unimore.it; agneseilaria.telloni@unimore.it)
Agnese Ilaria Telloni
Affiliation:
Dipartimento di Matematica, Università di Modena e Reggio Emilia, Via Campi 213/B, 41100 Modena, Italy (cavicchioli.alberto@unimore.it; spaggiari.fulvia@unimore.it; agneseilaria.telloni@unimore.it)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider orientable closed connected 3-manifolds obtained by performing Dehn surgery on the components of some classical links such as Borromean rings and twisted Whitehead links. We find geometric presentations of their fundamental groups and describe many of them as 2-fold branched coverings of the 3-sphere. Finally, we obtain some topological applications on the manifolds given by exceptional surgeries on hyperbolic 2-bridge knots.

Keywords

3-manifoldslinksDehn surgeryexceptional surgeriesgroup presentationscyclic branched coverings

MSC classification

Secondary: 57M12: Special coverings, e.g. branched 57R65: Surgery and handlebodies 20F05: Generators, relations, and presentations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 1 , February 2011 , pp. 33 - 45
Copyright
Copyright © Edinburgh Mathematical Society 2011

References

1.Birman, J. S. and Montesinos, J. M., On minimal Heegaard splittings, Michigan Math. J. 27 (1980), 47–57.CrossRefGoogle Scholar
2.Brittenham, M. and Wu, Y. Q., The classification of exceptional Dehn surgeries on 2-bridge knots, Commun. Analysis Geom. 9(1) (2001), 97–113.CrossRefGoogle Scholar
3.Burde, G. and Zieschang, H., Knots (Walter de Gruyter, Berlin, 1985).Google Scholar
4.Cavicchioli, A., Spaggiari, F. and Telloni, A. I., On the fundamental group of some surgery manifolds, in preparation.Google Scholar
5.Kirby, R., A calculus for framed links in , Invent. Math. 45 (1978), 35–56.CrossRefGoogle Scholar
6.Kirby, R., Problems in low-dimensional topology, in Geometric topology (ed. Kirby, R.), pp. 35–473 (American Mathematical Society, Providence, RI, 1997).Google Scholar
7.Lickorish, W. B. R., A representation of orientable combinatorial 3-manifolds, Annals Math. 76(2) (1962), 531–540.Google Scholar
8.Mednykh, A. D. and Vesnin, A. Yu., On Heegaard genus of three-dimensional hyperbolic manifolds of small volume, Sb. Math. J. 37(5) (1996), 893–897.Google Scholar
9.Mednykh, A. D. and Vesnin, A. Yu., Covering properties of small volume hyperbolic 3-manifolds, J. Knot Theory Ram. 7(3) (1998), 381–392.CrossRefGoogle Scholar
10.Montesinos, J. M., Variedades de Seifert que son recubridores ciclicos ramificados de dos hojas, Bol. Soc. Mat. Mex. 18 (1973), 1–32.Google Scholar
11.Montesinos, J. M., Surgery on links and double branched covers of , in Knots, groups and 3-manifolds (ed. L. P. Neuwirth), Annals of Mathematics Studies, Volume 84 (Princeton University Press, 1975).Google Scholar
12.Moser, L., Elementary surgery along a torus knot, Pac. J. Math. 38 (1971), 737–745.Google Scholar
13.Osborne, R. P. and Stevens, R. S., Group presentations corresponding to spines of 3-manifolds, I, Am. J. Math. 96 (1974), 454–471.Google Scholar
14.Osborne, R. P. and Stevens, R. S., Group presentations corresponding to spines of 3-manifolds, II, Trans. Am. Math. Soc. 234 (1977), 213–243.Google Scholar
15.Osborne, R. P. and Stevens, R. S., Group presentations corresponding to spines of 3-manifolds, III, Trans. Am. Math. Soc. 234 (1977), 245–251.Google Scholar
16.Thurston, W., Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Am. Math. Soc. 6 (1982), 357–381.CrossRefGoogle Scholar
17.Wallace, A. H., Modifications and cobounding manifolds, Can. J. Math. 12 (1960), 503–528.CrossRefGoogle Scholar
18.Weeks, J. R., Computer program SnapPea and tables of volumes and isometries of knots, links, and manifolds (available by anonymous ftp from http:\\www.geometrygames.orgindirectorypub/software/snappea/).Google Scholar