Mutation on Knots and Whitney＇s 2-Isomorphism Theorem
Whitney＇s 2-switching theorem states that any two embeddings of a 2-connected planar graph in S2 can be connected via a sequence of simple operations, named 2-switching. In this paper, we obtain two operations on planar graphs from the view point of knot theory, which we will term ＂twisting＂ and ＂2-...
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| Published in | 数学学报：英文版 no. 6; pp. 1219 - 1230 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
2013
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| Subjects | |
| Online Access | Get full text |
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| Abstract | Whitney＇s 2-switching theorem states that any two embeddings of a 2-connected planar graph in S2 can be connected via a sequence of simple operations, named 2-switching. In this paper, we obtain two operations on planar graphs from the view point of knot theory, which we will term ＂twisting＂ and ＂2-switching＂ respectively. With the twisting operation, we give a pure geometrical proof of Whitney＇s 2-switching theorem. As an application, we obtain some relationships between two knots which correspond to the same signed planar graph. Besides, we also give a necessary and sufficient condition to test whether a pair of reduced alternating diagrams are mutants of each other by their signed planar graphs. |
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| AbstractList | Whitney＇s 2-switching theorem states that any two embeddings of a 2-connected planar graph in S2 can be connected via a sequence of simple operations, named 2-switching. In this paper, we obtain two operations on planar graphs from the view point of knot theory, which we will term ＂twisting＂ and ＂2-switching＂ respectively. With the twisting operation, we give a pure geometrical proof of Whitney＇s 2-switching theorem. As an application, we obtain some relationships between two knots which correspond to the same signed planar graph. Besides, we also give a necessary and sufficient condition to test whether a pair of reduced alternating diagrams are mutants of each other by their signed planar graphs. |
| Author | Zhi Yun CHENG Hong Zhu GAO |
| AuthorAffiliation | School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, P. R. China |
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| DocumentTitleAlternate | Mutation on Knots and Whitney＇s 2-Isomorphism Theorem |
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| Notes | Planar graph, twisting, 2-switching, mutation Whitney＇s 2-switching theorem states that any two embeddings of a 2-connected planar graph in S2 can be connected via a sequence of simple operations, named 2-switching. In this paper, we obtain two operations on planar graphs from the view point of knot theory, which we will term ＂twisting＂ and ＂2-switching＂ respectively. With the twisting operation, we give a pure geometrical proof of Whitney＇s 2-switching theorem. As an application, we obtain some relationships between two knots which correspond to the same signed planar graph. Besides, we also give a necessary and sufficient condition to test whether a pair of reduced alternating diagrams are mutants of each other by their signed planar graphs. 11-2039/O1 |
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| PublicationDate | 2013 |
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| PublicationTitle | 数学学报：英文版 |
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| PublicationYear | 2013 |
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| Snippet | Whitney＇s 2-switching theorem states that any two embeddings of a 2-connected planar graph in S2 can be connected via a sequence of simple operations, named... |
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| StartPage | 1219 |
| SubjectTerms | 充分必要条件 切换 同构定理 嵌入定理 平面图形 操作 突变 纽结理论 |
| Title | Mutation on Knots and Whitney＇s 2-Isomorphism Theorem |
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