Abstract
Whitney’s 2-switching theorem states that any two embeddings of a 2-connected planar graph in S 2 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Xu, J. M.: Theory and Application of Graphs, Kluwer Academic Publishers, Dordrechet-Boston-London, 2003
Graham, R. L., Grotschel, M., Lovasz, L.: Handbook of Combinatorics (Vol. 1), MIT Press, Massachusetts, 1996
Whitney, H.: 2-Isomorphic graphs. Amer. Math. J., 55, 245–254 (1933)
Adams, C. C.: The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, W. H. Freeman and Company, New York, 2004
Gross, J. L., Yellen, J.: Graph Theory and Its Applications, Second Edition, Chapman & Hall/CRC, New York, 2006
Truemper, K.: On Whitney’s 2-isomorphism theorem for graphs. J. Graph Theory, 4(1) 43–49 (1980)
Thomassen, C.: Whitney’s 2-switching theorem, cycle space, and arc mappings of directed graphs. J. Comb. Theory Ser. B, 46(3), 257–291 (1989)
Mohar, B.: Combinatorial local planarity and the width of graph embeddings. Canad. J. Math., 44(6), 1272–1288 (1992)
Freyd, P., Yetter, D., Hoste, J., et al.: A new polynomial invariant of knots and links. Bull. Amer. Math. Soc., 12(2), 239–246 (1985)
Ruberman, D.: Mutation and volumes of knots in S 3. Invent. Math., 90, 189–215 (1987)
Champanerkar, A., Kofman, I.: On mutation and Khovanov homology. Communications in Contemporary Mathematics, 10, 973–992 (2008)
Menasco, W., Thistlethwaite, M.: The Tait flyping conjecture. Bull. Amer. Math. Soc., 25(2), 403–412 (1991)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by National Natural Science Foundation of China (No. 11171025) and Science Foundation for the Youth Scholars of Beijing Normal University
Electronic supplementary material
Rights and permissions
About this article
Cite this article
Cheng, Z.Y., Gao, H.Z. Mutation on knots and Whitney’s 2-isomorphism theorem. Acta. Math. Sin.-English Ser. 29, 1219–1230 (2013). https://doi.org/10.1007/s10114-013-0679-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-013-0679-5