Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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EVERY LINK IS A BOUNDARY OF A COMPLETE BIPARTITE GRAPH K2,n

  • Received : 2012.08.21
  • Accepted : 2012.11.05
  • Published : 2012.12.30
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Abstract

A voltage assignment on a graph was used to enumerate all possible 2-cell embeddings of a graph onto surfaces. The boundary of the surface which is obtained from 0 voltage on every edges of a very special diagram of a complete bipartite graph $K_{m,n}$ is surprisingly the ($m,n$) torus link. In the present article, we prove that every link is the boundary of a complete bipartite multi-graph $K_{m,n}$ for which voltage assignments are either -1 or 1 and that every link is the boundary of a complete bipartite graph $K_{2,n}$ for which voltage assignments are either -1, 0 or 1 where edges in the diagram of graphs may be linked but not knotted.

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References

  1. C. Adams, The knot book, W.H. Freeman and Company. 1994.
  2. S. Baader, Bipartite graphs and combinatorial adjacency, preprint, arXiv:1111.3747.
  3. R. Furihata, M. Hirasawa and T. Kobayashi, Seifert surfaces in open books, and a new coding algorithm for links, Bull. Lond. Math. Soc. 40 (3) (2008), 405-414. https://doi.org/10.1112/blms/bdn020
  4. D. Gabai, Genera of the arborescent links, Mem. Amer. Math. Soc. 59 (339) (1986) I.VIII, 1-98.
  5. J. Gross and T. Tucker, Topological graph theory, Wiley-Interscience Series in discrete Mathematics and Optimization, Wiley & Sons, New York, 1987.
  6. D. Kim, Basket, flat plumbing and flat plumbing basket surfaces derived from induced graphs, preprint, arXiv:1108.1455.
  7. D. Kim, Y.S. Kwon and J. Lee, String surfaces, string indexes and genera of links, preprint, arXiv:1105.0059.
  8. L. Rudolph, Braided surfaces and Seifert ribbons for closed braids, Comment. Math. Helv. 58 (1) (1983) 1-37. https://doi.org/10.1007/BF02564622
  9. L. Rudolph, Hopf plumbing, arborescent Seifert surfaces, baskets, espaliers, and homogeneous braids, Topology Appl. 116 (2001), 255-277. https://doi.org/10.1016/S0166-8641(00)90091-9
  10. H. Seifert, Uber das Geschlecht von Knoten, Math. Ann. 110 (1934), 571-592.
  11. J. Stallings, Constructions of fibred knots and links, in: Algebraic and Geometric Topology (Proc. Sympos. PureMath., Stanford Univ., Stanford, CA, 1976), Part 2, Amer. Math. Soc., Providence, RI, 1978, pp. 55-60.
  12. T. Van Zandt. PSTricks: PostScript macros for generic TEX. Available at ftp://ftp. princeton.edu/pub/tvz/.

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