%0 Journal Article
%T The Dual and Mirror Images of the Dunwoody 3-Manifolds
%A Soo Hwan Kim
%A Yangkok Kim
%J International Journal of Mathematics and Mathematical Sciences
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/103209
%X Recently, in 2013, we proved that certain presentations present the Dunwoody -manifold groups. Since the Dunwoody -manifolds do not have a unique Heegaard diagram, we cannot determine a unique group presentation for the Dunwoody -manifolds. It is well known that every -knots in a lens space can be represented by the set of the 4-tuples (Cattabriga and Mulazzani (2004); S. H. Kim and Y. Kim (2012, 2013)). In particular, to determine a unique Heegaard diagram of the Dunwoody -manifolds, we proved the fact that the certain subset of representing all -bridge knots of -knots is determined completely by using the dual and mirror -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). 1. Introduction In the study of the Dunwoody -manifolds, Heegaard splittings and Heegaard diagrams provide a simple means of understanding the Dunwoody -manifolds by changing a -dimensional problem into a -dimensional problem, for the basic definitions of Heegaard splittings and Heegaard diagrams; see [1, 2]. Even with the help of Heegaard diagrams, because the Dunwoody -manifolds do not have a unique Heegaard diagram, generally, the following are still open questions about -manifolds: given -manifolds and , (homeomorphism problem) is ? or (isomorphism problem) is ? The Dunwoody -manifolds have a Heegaard diagram, from which one can obtain presentations for a group; however, not all group presentations arise from Heegaard diagrams of Dunwoody -manifolds. The Heegaard diagrams are useful for understanding properties of the Dunwoody -manifolds as there is a correspondence between the Heegaard diagrams and the fundamental groups for the Dunwoody -manifolds, allowing transformations of a group presentation for the fundamental groups to a simple calculus of the Heegaard diagrams. Thus, transformations of the Heegaard diagrams are corresponding to transformations of a group presentation for the fundamental groups of the Dunwoody -manifolds. In [3], we gave some conditions for the answer to the generalized Dunwoody -manifolds, constructed by group presentations for the fundamental group. For an answer to the above problems between the Dunwoody -manifolds, we consider the dual and mirror images of the Dunwoody -manifolds and give the basic definitions about them in the following. Let be a Heegaard splitting of a -manifold with genus . A properly embedded disc in the handlebody is called a meridian disc of if
%U http://www.hindawi.com/journals/ijmms/2013/103209/