{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T10:41:49Z","timestamp":1740134509497,"version":"3.37.3"},"reference-count":18,"publisher":"Wiley","license":[{"start":{"date-parts":[[2013,1,1]],"date-time":"2013-01-01T00:00:00Z","timestamp":1356998400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100002469","name":"Dong-Eui University","doi-asserted-by":"publisher","award":["2012AA085"],"award-info":[{"award-number":["2012AA085"]}],"id":[{"id":"10.13039\/501100002469","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2013]]},"abstract":"Recently, in 2013, we proved that certain presentations present the Dunwoody3<\/mml:mn><\/mml:mrow><\/mml:math>-manifold groups. Since the Dunwoody3<\/mml:mn><\/mml:mrow><\/mml:math>-manifolds do not have a unique Heegaard diagram, we cannot determine a unique group presentation for the Dunwoody3<\/mml:mn><\/mml:mrow><\/mml:math>-manifolds. It is well known that every(<\/mml:mo>1,1<\/mml:mn>)<\/mml:mo><\/mml:math>-knots in a lens space can be represented by the set\ud835\udc9f<\/mml:mi><\/mml:mrow><\/mml:math>of the 4-tuples(<\/mml:mo>a<\/mml:mi>,<\/mml:mo>b<\/mml:mi>,<\/mml:mo>c<\/mml:mi>,<\/mml:mo>r<\/mml:mi>)<\/mml:mo><\/mml:math>(Cattabriga and Mulazzani (2004); S. H. Kim and Y. Kim (2012, 2013)). In particular, to determine a unique Heegaard diagram of the Dunwoody3<\/mml:mn><\/mml:mrow><\/mml:math>-manifolds, we proved the fact that the certain subset of\ud835\udc9f<\/mml:mi><\/mml:mrow><\/mml:math>representing all2<\/mml:mn><\/mml:mrow><\/mml:math>-bridge knots of(<\/mml:mo>1,1<\/mml:mn>)<\/mml:mo><\/mml:math>-knots is determined completely by using the dual and mirror(<\/mml:mo>1,1<\/mml:mn>)<\/mml:mo><\/mml:math>-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\ud835\udc9f<\/mml:mi><\/mml:mrow><\/mml:math>as the generalization of some results by Grasselli and Mulazzani (2001); S. H. Kim and Y. Kim (2011).<\/jats:p>","DOI":"10.1155\/2013\/103209","type":"journal-article","created":{"date-parts":[[2013,9,18]],"date-time":"2013-09-18T17:01:33Z","timestamp":1379523693000},"page":"1-7","source":"Crossref","is-referenced-by-count":0,"title":["The Dual and Mirror Images of the Dunwoody 3-Manifolds"],"prefix":"10.1155","volume":"2013","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5213-9213","authenticated-orcid":true,"given":"Soo Hwan","family":"Kim","sequence":"first","affiliation":[{"name":"Department of Mathematics, Dong-eui University, Busan 614-714, Republic of Korea"}]},{"given":"Yangkok","family":"Kim","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Dong-eui University, Busan 614-714, Republic of Korea"}]}],"member":"311","reference":[{"volume":"86","volume-title":"3-manifolds","year":"1976","key":"9"},{"year":"1978","key":"18"},{"issue":"2","key":"15","first-page":"457","volume":"50","year":"2013","journal-title":"Osaka Journal of Mathematics"},{"first-page":"47","volume-title":"Cyclic resentations and (1,1)-manifolds","year":"1995","key":"6"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004103006686"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1112\/S0024610704005538"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1515\/form.2001.013"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.5666\/KMJ.2012.52.2.223"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.4134\/BKMS.2011.48.1.197"},{"key":"1","doi-asserted-by":"publisher","DOI":"10.1023\/B:SIMJ.0000013008.25556.29"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1142\/S0218216504003603"},{"key":"16","doi-asserted-by":"publisher","DOI":"10.4134\/BKMS.2003.40.1.101"},{"key":"17","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1017\/S0305004100043279","volume":"64","year":"1968","journal-title":"Proceedings of the Cambridge Philosophical Society"},{"issue":"1","key":"10","first-page":"111","volume":"23","year":"2004","journal-title":"Note di Matematica"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1515\/advg.2004.016"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.4064\/fm188-0-3"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1007\/s11202-009-0003-x"},{"issue":"1","key":"11","first-page":"199","volume":"6","year":"2003","journal-title":"Bollettino della Unione Matematica Italiana"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2013\/103209.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2013\/103209.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2013\/103209.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,21]],"date-time":"2017-06-21T18:08:06Z","timestamp":1498068486000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/ijmms\/2013\/103209\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013]]},"references-count":18,"alternative-id":["103209","103209"],"URL":"https:\/\/doi.org\/10.1155\/2013\/103209","relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"type":"print","value":"0161-1712"},{"type":"electronic","value":"1687-0425"}],"subject":[],"published":{"date-parts":[[2013]]}}}