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THE BOUNDARIES OF DIPOLE GRAPHS AND THE COMPLETE BIPARTITE GRAPHS K2,n

  • Kim, Dongseok (Department of Mathematics, Kyonggi University)
  • Received : 2014.03.15
  • Accepted : 2014.04.11
  • Published : 2014.06.25

Abstract

We study the Seifert surfaces of a link by relating the embeddings of graphs with induced graphs. As applications, we prove that every link L is the boundary of an oriented surface which is obtained from a graph embedding of a complete bipartite graph $K_{2,n}$, where all voltage assignments on the edges of $K_{2,n}$ are 0. We also provide an algorithm to construct such a graph diagram of a given link and demonstrate the algorithm by dealing with the links $4^2_1$ and $5_2$.

Keywords

Seifert surfaces;links;graph embeddings;dipoles;complete bopartite graphs

Acknowledgement

Supported by : Kyonggi University

References

  1. V. Krushkal, Graphs, links and duality on surfaces, Comb. Probab. Comput. 20 (2011), 267-287. https://doi.org/10.1017/S0963548310000295
  2. L. Kauffman, On Knots, The Annals of Mathematics Studies, Princeton University Press, New Jersey, 1987.
  3. L. Kauffman, State models and the Jones polynomial, Topology 26(3) (1987), 395-407. https://doi.org/10.1016/0040-9383(87)90009-7
  4. D. Kim, Basket, flat plumbing and flat plumbing basket surfaces derived from induced graphs, preprint, arXiv:1108.1455.
  5. D. Kim, Y. S. Kwon and J. Lee, Banded surfaces, banded links, band indices and genera of links, J. Knot Theory Ramifications, 22(7) (2013), 1350035, 1-18.
  6. D. Kim, Y. S. Kwon and J. Lee, The existence of an alternating sign on a spanning tree of graphs, Kyungpook Math. J. 52(4) (2012), 513-519. https://doi.org/10.5666/KMJ.2012.52.4.513
  7. T. Miura, On flat braidzel surface for links, Topology Appl. 159 (2012), 623-632. https://doi.org/10.1016/j.topol.2011.10.009
  8. T. Nakamura, Notes on braidzel surfaces for links, Proc. of AMS 135(2) (2007), 559-567. https://doi.org/10.1090/S0002-9939-06-08478-4
  9. H. Seifert, Uber das Geschlecht von Knoten, Math. Ann. 110 (1934), 571-592.
  10. M. Thistlethwaite, A spanning tree expansion of the Jones polynomial, Topology 26(3) (1987), 297-309. https://doi.org/10.1016/0040-9383(87)90003-6
  11. T. Van Zandt. PSTricks: PostScript macros for generic TEX. Available at ftp://ftp.princeton.edu/pub/tvz/.
  12. T. Visentin and S.Wieler, On the genus distribution of (p, q, n)-dipoles, Electron. J. Comb. 14 (2007), #12.
  13. S. Wehrli, A spanning tree model for Khovanov homology, J. Knot Theory Ramifications, 17(12) (2008), 1561-1574. https://doi.org/10.1142/S0218216508006762
  14. S. Baader, Bipartite graphs and combinatorial adjacency, preprint, arXiv:1111.3747.
  15. S. Bang, D. Kim, Y. S. Kwon and J. Lee, Graphs, links and surfaces, preprint.
  16. G. Burde and H. Zieschang, Knots, Berlin, de Gruyter, 1985.
  17. O. Dasbach, D. Futer, E. Kalfagianni, X-S Lin, N. Stoltzfus, The Jones polyno-mial and graphs on surfaces, J. Combin. Theory Ser. B 98(2) (2008), 384-399. https://doi.org/10.1016/j.jctb.2007.08.003
  18. R. Furihata, M. Hirasawa and T. Kobayashi, Seifert surfaces in open books, and a new coding algorithm for links, Bull. London Math. Soc. 40(3) (2008), 405-414. https://doi.org/10.1112/blms/bdn020
  19. J. Gross and T. Tucker, Topological graph theory, Wiley-Interscience Series in discrete Mathematics and Optimization, Wiley & Sons, New York, 1987.
  20. Y. Jang, S. Jeon and D. Kim, Every link is a boundary of a complete bipartite graph $K_{2,n}$, Korean J. of Math. 20(4) (2012), 403-414. https://doi.org/10.11568/kjm.2012.20.4.403

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