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
Main Author Zhi Yun CHENG Hong Zhu GAO
Format Journal Article
LanguageEnglish
Published 2013
Subjects
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
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Summary: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.
Bibliography: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
ISSN:1439-8516
1439-7617