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Genus one 1-bridge knots and Dunwoody manifolds

Genus one 1-bridge knots and Dunwoody manifolds Abstract. In this paper we show that all 3-manifolds of a family introduced by M. J. Dunwoody are cyclic coverings of lens spaces (possibly S 3 ), branched over genus one 1-bridge knots. As a consequence, we give a positive answer to the Dunwoody conjecture that all the elements of a wide subclass are cyclic coverings of S 3 branched over a knot. Moreover, we show that all branched cyclic coverings of a 2-bridge knot belong to this subclass; this implies that the fundamental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation. 1991 Mathematics Subject Classi®cation: 57M12, 57M25; 20F05, 57M05. 1 Introduction and preliminaries The problem of determining if a balanced presentation of a group is geometric (i.e. induced by a Heegaard diagram of a closed orientable 3-manifold) is quite important within geometric topology and has been deeply investigated by many authors (see [9], [22], [25], [26], [27], [28], [33]); further, the connections between branched cyclic coverings of links and cyclic presentations of groups induced by suitable Heegaard diagrams have been recently pointed out in several papers (see [1], [3], [4], [6], [11], [12], [16], [18], [17], [20], [34]). In order http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Genus one 1-bridge knots and Dunwoody manifolds

Grasselli, Luigi; Mulazzani, Michele
Forum Mathematicum , Volume 13 (3) – Apr 5, 2001
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Publisher
de Gruyter
Copyright
Copyright © 2001 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.2001.013
Publisher site
See Article on Publisher Site

Abstract

Abstract. In this paper we show that all 3-manifolds of a family introduced by M. J. Dunwoody are cyclic coverings of lens spaces (possibly S 3 ), branched over genus one 1-bridge knots. As a consequence, we give a positive answer to the Dunwoody conjecture that all the elements of a wide subclass are cyclic coverings of S 3 branched over a knot. Moreover, we show that all branched cyclic coverings of a 2-bridge knot belong to this subclass; this implies that the fundamental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation. 1991 Mathematics Subject Classi®cation: 57M12, 57M25; 20F05, 57M05. 1 Introduction and preliminaries The problem of determining if a balanced presentation of a group is geometric (i.e. induced by a Heegaard diagram of a closed orientable 3-manifold) is quite important within geometric topology and has been deeply investigated by many authors (see [9], [22], [25], [26], [27], [28], [33]); further, the connections between branched cyclic coverings of links and cyclic presentations of groups induced by suitable Heegaard diagrams have been recently pointed out in several papers (see [1], [3], [4], [6], [11], [12], [16], [18], [17], [20], [34]). In order

Journal

Forum Mathematicum – de Gruyter

Published: Apr 5, 2001

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