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...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2013-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2013/103209 |
| Summary: | 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). |
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| ISSN: | 0161-1712 1687-0425 |