Genus one 1-bridge knots and Dunwoody manifolds

Grasselli, Luigi; Mulazzani, Michele

doi:10.1515/form.2001.013

Cited by 26 publications

(37 citation statements)

References 45 publications

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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%

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Strongly-cyclic branched coverings of (1, 1)-knots and cyclic presentations of groups

Cattabriga¹,

Mulazzani²

2003

Math. Proc. Camb. Phil. Soc.

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We study the connections among the mapping class group of the twice punctured torus, the cyclic branched coverings of (1, 1)-knots and the cyclic presentations of groups. We give the necessary and sufficient conditions for the existence and uniqueness of the n-fold stronglycyclic branched coverings of (1, 1)-knots, through the elements of the mapping class group. We prove that every n-fold strongly-cyclic branched covering of a (1, 1)-knot admits a cyclic presentation for the fundamental group, arising from a Heegaard splitting of genus n. Moreover, we give an algorithm to produce the cyclic presentation and illustrate it in the case of cyclic branched coverings of torus knots of type (k, hk ± 1).

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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%

Strongly-cyclic branched coverings of (1, 1)-knots and cyclic presentations of groups

Cattabriga¹,

Mulazzani²

2003

Math. Proc. Camb. Phil. Soc.

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“…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%

Representations of (1,1)-knots

Cattabriga

Mulazzani

2005

Fund. Math.

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Abstract. We present two different representations of (1, 1)-knots and study some connections between them. The first representation is algebraic: every (1, 1)-knot is represented by an element of the pure mapping class group of the twice punctured torus PMCG 2 (T ). Moreover, there is a surjective map from the kernel of the natural homomorphism Ω : PMCG 2 (T ) → MCG(T ) ∼ = SL(2, Z), which is a free group of rank two, to the class of all (1, 1)-knots in a fixed lens space. The second representation is parametric: every (1, 1)-knot can be represented by a 4-tuple (a, b, c, r) of integer parameters such that a, b, c ≥ 0 and r ∈ Z 2a+b+c . The strict connection of this representation with the class of Dunwoody manifolds is illustrated. The above representations are explicitly obtained in some interesting cases, including two-bridge knots and torus knots.

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“…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%

Parameterizations of 1-Bridge Torus Knots

Choi

2003

J. Knot Theory Ramifications

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Abstract. A 1-bridge torus knot in a 3-manifold of genus ≤ 1 is a knot drawn on a Heegaard torus with one bridge. We give two types of normal forms to parameterize the family of 1-bridge torus knots that are similar to the Schubert's normal form and the Conway's normal form for 2-bridge knots. For a given Schubert's normal form we give algorithms to determine the number of components and to compute the fundamental group of the complement when the normal form determines a knot. We also give a description of the double branched cover of an ambient 3-manifold branched along a 1-bridge torus knot by using its Conway's normal form and obtain an explicit formula for the first homology of the double cover.

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Genus one 1-bridge knots and Dunwoody manifolds

Cited by 26 publications

References 45 publications

Strongly-cyclic branched coverings of (1, 1)-knots and cyclic presentations of groups

Strongly-cyclic branched coverings of (1, 1)-knots and cyclic presentations of groups

Representations of (1,1)-knots

Parameterizations of 1-Bridge Torus Knots

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