@article{oai:ynu.repo.nii.ac.jp:00006955,
 author = {Nakamoto, Atsuhiro},
 issue = {Special},
 journal = {Yokohama Mathematical Journal = 横濱市立大學紀要. D部門, 数学},
 month = {},
 note = {application/pdf, Let ＄G＄ be an embedding on the torus and let ＄￥ell＄ be an essential closed curve on the torus. Let ＄f_{G}(￥ell)＄ be the minimum number of intersections of ＄G＄ and a ＄c1_{08}ed＄ curve ＄￥ell^{￥prime}＄ , where ＄￥ell^{￥prime}＄ ranges over all closed curves homotopic to ＄￥ell＄ . An embedding ＄G＄ on the torus is said to be akernel if for any proper minor ＄T＄ of ＄G＄ , ＄f_{G}(￥ell)>f_{T}(￥ell)＄ for some ＄￥ell＄ . In this paper, we show that any two kernels ＄G＄ and ＄G^{￥prime}＄ on the torus with ＄f_{G}=f_{G}＄, can be transformed into each other by a sequence of ＄Y￥Delta-e＄xchanges.},
 pages = {183--189},
 title = {THE ＄ Y￥Delta＄ -EQUIVALENCE OF KERNELS ON THE TORUS WITHOUT TAKING DUALS},
 volume = {47},
 year = {1999}
}