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...
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| Format: | Article |
| Language: | English |
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Wiley
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 |
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| author | Soo Hwan Kim Yangkok Kim |
| author_facet | Soo Hwan Kim Yangkok Kim |
| author_sort | Soo Hwan Kim |
| collection | DOAJ |
| description | 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). |
| format | Article |
| id | doaj-art-34c738658e454675a4a2896f30220041 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-34c738658e454675a4a2896f302200412025-02-03T05:44:51ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252013-01-01201310.1155/2013/103209103209The Dual and Mirror Images of the Dunwoody 3-ManifoldsSoo Hwan Kim0Yangkok Kim1Department of Mathematics, Dong-eui University, Busan 614-714, Republic of KoreaDepartment of Mathematics, Dong-eui University, Busan 614-714, Republic of KoreaRecently, 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).http://dx.doi.org/10.1155/2013/103209 |
| spellingShingle | Soo Hwan Kim Yangkok Kim The Dual and Mirror Images of the Dunwoody 3-Manifolds International Journal of Mathematics and Mathematical Sciences |
| title | The Dual and Mirror Images of the Dunwoody 3-Manifolds |
| title_full | The Dual and Mirror Images of the Dunwoody 3-Manifolds |
| title_fullStr | The Dual and Mirror Images of the Dunwoody 3-Manifolds |
| title_full_unstemmed | The Dual and Mirror Images of the Dunwoody 3-Manifolds |
| title_short | The Dual and Mirror Images of the Dunwoody 3-Manifolds |
| title_sort | dual and mirror images of the dunwoody 3 manifolds |
| url | http://dx.doi.org/10.1155/2013/103209 |
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