The Dual and Mirror Images of the Dunwoody 3-Manifolds - Soo Hwan Kim; Yangkok Kim (International Journal of Mathematics and Mathematical Sciences)
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 𝒟 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 𝒟 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 𝒟 as the generalization of some results by Grasselli and Mulazzani (2001); S. H. Kim and Y. Kim (2011).