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-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.