On Hyperbolic 3-Manifolds Obtained by Dehn Surgery on Links

We study the algebraic and geometric structures for closed orientable 3-manifolds obtained by Dehn surgery along the family of hyperbolic links with certain surgery coefficients and moreover, the geometric presentations of the fundamental group of these manifolds. We prove that our surgery manifolds are 2-fold cyclic covering of 3-sphere branched over certain link by applying the Montesinos theorem in Montesinos-Amilibia (1975). In particular, our result includes the topological classification of the closed 3-manifolds obtained by Dehn surgery on the Whitehead link, according to Mednykh and Vesnin (1998), and the hyperbolic link +1 of +1 components in Cavicchioli and Paoluzzi (2000)

About some infinite family of 2-bridge knots and 3-manifolds

We construct an infinite family of 3-manifolds and show that these manifolds have cyclically presented fundamental groups and are cyclic branched coverings of the 3-sphere branched over the 2-bridge knots (ℓ+1)2 or (ℓ+1)1, that are the closure of the rational (2ℓ−1)/(ℓ−1)-tangles or (2ℓ−1)/ℓ-tangles, respectively

The Dual and Mirror Images of the Dunwoody 3-Manifolds

Recently, in 2013, we proved that certain presentations present the Dunwoody 3-manifold groups. Since the Dunwoody 3-manifolds do not have a unique Heegaard diagram, we cannot determine a unique group presentation for the Dunwoody 3-manifolds. It is well known that every (1,1)-knots in a lens space can be represented by the set of 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 Dunwoody 3-manifolds, we proved the fact that the certain subset of representing all 2-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 in as the generalization of some results by Grasselli and Mulazzani (2001); S. H. Kim and Y. Kim (2011)