Seifert manifolds and (1, 1)-knots

Grasselli, Luigi; Mulazzani, Michele

doi:10.1007/s11202-009-0003-x

Cited by 4 publications

(4 citation statements)

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“…Proof. By results of [10], we have that S n (p, q, ℓ) = M(q, q(nℓ − 2), p − 2q, n, p − q, 0) if p 2q and S n (p, q, ℓ) = M(p − q, 2q − p, q(nℓ − 2), n, p − q, 1) otherwise. The result follows from Proposition 7.…”

Section: Dunwoody Manifoldsmentioning

confidence: 99%

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Complexity, Heegaard Diagrams and Generalized Dunwoody Manifolds

Cattabriga¹,

Mulazzani²,

Vesnin³

2010

Journal of the Korean Mathematical Society

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Abstract. We deal with Matveev complexity of compact orientable 3-manifolds represented via Heegaard diagrams. This lead us to the definition of modified Heegaard complexity of Heegaard diagrams and of manifolds. We define a class of manifolds which are generalizations of Dunwoody manifolds, including cyclic branched coverings of two-bridge knots and links, torus knots, some pretzel knots, and some theta-graphs. Using modified Heegaard complexity, we obtain upper bounds for their Matveev complexity, which linearly depend on the order of the covering. Moreover, using homology arguments due to Matveev and Pervova we obtain lower bounds.

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Section: Dunwoody Manifoldsmentioning

confidence: 99%

“…It is proved in [10] that if p > q > 0 and gcd(p, q) = 1, n > 1, ℓ > 0, then Seifert manifolds S n (p, q, ℓ) = {Oo, 0 | −1; (p, q), . .…”

Section: Dunwoody Manifoldsmentioning

confidence: 99%

Complexity, Heegaard Diagrams and Generalized Dunwoody Manifolds

Cattabriga¹,

Mulazzani²,

Vesnin³

2010

Journal of the Korean Mathematical Society

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“…In addition to these very general presentations, there are a number of concrete special cases in the literature [Bleiler and Moriah 1988;Kim 2000;Kim and Kim 2003;Jeong 2006;Jeong and Wang 2008;Grasselli and Mulazzani 2009;Telloni 2010].…”

Section: Ementioning

confidence: 99%

Untitled

2016

Pacific J. Math.

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Though LeeR Lambert spent his life as an actuary and a musician and was a loving father of nine girls and one boy, he had always wanted to earn an advanced degree as a mathematician. With the encouragement of his wife, he earned his Ph.D. in mathematics at the age of 68. Many of the results of this paper appeared in his Ph.D. dissertation at Brigham Young University. At the age of 71, LeeR died of bone cancer. We miss you, LeeR. We give uniform, explicit, and simple face-pairing descriptions of all the branched cyclic covers of the 3-sphere, branched over two-bridge knots. Our method is to use the bitwisted face-pairing constructions of Cannon, Floyd, and Parry; these examples show that the bitwist construction is often efficient and natural. Finally, we give applications to computations of fundamental groups and homology of these branched cyclic covers.

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“…Therefore, the concept of the Dunwoody 3-manifolds is important in knots, branched coverings, and graph theories. Moreover, recent manuscripts of the Dunwoody 3-manifolds can be found in [3,8,10,12,[16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

The Dual and Mirror Images of the Dunwoody 3-Manifolds

Kim

2013

International Journal of Mathematics and Mathematical Sciences

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Recently, in 2013, we proved that certain presentations present the Dunwoody3-manifold groups. Since the Dunwoody3-manifolds do not have a unique Heegaard diagram, we cannot determine a unique group presentation for the Dunwoody3-manifolds. It is well known that every(1,1)-knots in a lens space can be represented by the set𝒟of the 4-tuples(a,b,c,r)(Cattabriga and Mulazzani (2004); S. H. Kim and Y. Kim (2012, 2013)). In particular, to determine a unique Heegaard diagram of the Dunwoody3-manifolds, we proved the fact that the certain subset of𝒟representing all2-bridge knots of(1,1)-knots is determined completely by using the dual and mirror(1,1)-decompositions (S. H. Kim and Y. Kim (2011)). In this paper, we show how to obtain the dual and mirror images of all elements in𝒟as the generalization of some results by Grasselli and Mulazzani (2001); S. H. Kim and Y. Kim (2011).

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Seifert manifolds and (1, 1)-knots

Abstract: The aim of this paper is to investigate the relations between Seifert manifolds and (1, 1)-knots. In particular, we prove that each orientable Seifert manifold with invariants

Cited by 4 publications

References 28 publications

Complexity, Heegaard Diagrams and Generalized Dunwoody Manifolds

Complexity, Heegaard Diagrams and Generalized Dunwoody Manifolds

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The Dual and Mirror Images of the Dunwoody 3-Manifolds

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