2001
DOI: 10.1515/form.2001.013
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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 (eventually 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… Show more

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Cited by 26 publications
(37 citation statements)
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“…A generator Ψ of the group of covering transformations cyclically permutes these components. Let E ′ be a meridian disk for the torus We remark that the previous result for 2-bridge knots has also been obtained (by another technique) in [11] and the result for torus knots largely generalizes a result obtained in [5] and [6].…”
Section: Connections With Cyclic Presentations Of Groupssupporting
confidence: 73%
See 1 more Smart Citation
“…A generator Ψ of the group of covering transformations cyclically permutes these components. Let E ′ be a meridian disk for the torus We remark that the previous result for 2-bridge knots has also been obtained (by another technique) in [11] and the result for torus knots largely generalizes a result obtained in [5] and [6].…”
Section: Connections With Cyclic Presentations Of Groupssupporting
confidence: 73%
“…In [11] it has been shown that the 3-manifolds represented by these diagrams -the so-called Dunwoody manifolds -are cyclic coverings of lens spaces, branched over (1, 1)-knots. As a corollary, it has been proved that for some well-determined cases the manifolds turn out to be cyclic coverings of S 3 , branched over knots.…”
Section: Introductionmentioning
confidence: 99%
“…For example, the 4-tuples (a, 0, a, a), with a > 1, and (1, 0, c, 2), with c even, do not determine any (1, 1)-knot (see [10]). …”
Section: Proposition 3 ([6]) the Two-bridge Knot Having Conway Parammentioning
confidence: 99%
“…In [7], a subfamily of 1-bridge torus knots is completely classified using their genera and Jones polynomial. Also cyclic branched covers of ambient spaces along 1-bridge torus knots are known as Dunwoody 3-manifolds and are studied in [24] and [13].…”
Section: Introductionmentioning
confidence: 99%