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International Journal of Mathematics and Mathematical Sciences
Volume 2013 (2013), Article ID 103209, 7 pages
Research Article

The Dual and Mirror Images of the Dunwoody 3-Manifolds

Department of Mathematics, Dong-eui University, Busan 614-714, Republic of Korea

Received 1 April 2013; Revised 1 July 2013; Accepted 22 July 2013

Academic Editor: Seppo Hassi

Copyright © 2013 Soo Hwan Kim and Yangkok Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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).