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 pla...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.展开更多
基金Supported by National Natural Science Foundation of China(No.11171025)Science Foundation for the Youth Scholars of Beijing Normal University
文摘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.
