Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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The Kauman Polynomial and Trivalent Graphs

  • CAPRAU, CARMEN (Department of Mathematics, California State University-Fresno) ;
  • TIPTON, JAMES (Department of Mathematics, The University of Iowa)
  • Received : 2012.10.05
  • Accepted : 2013.04.04
  • Published : 2015.12.23
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Abstract

We construct a state model for the two-variable Kauman polynomial using planar trivalent graphs. We also use this model to obtain a polynomial invariant for a certain type of trivalent graphs embedded in $\mathbb{R}^3$.

Keywords

References

  1. J. W. Alexander, A lemma on systems of knotted curves, Proc. Nat. Acad. Sci., USA, 9(1923), 93-95.
  2. J. S. Birman, Braids, links and mapping class groups, Annals of Math Study, no 82. Princeton University Press (1974).
  3. J. S. Birman and H. Wenzl, Braids, link polynomial and a new algebra, Trans. Amer. Math. Soc., 313(1)(1989), 249-273. https://doi.org/10.1090/S0002-9947-1989-0992598-X
  4. R. P. Carpentier, From planar graphs to embedded graphs - a new approach to Kauffman and Vogel's polynomial, J. Knot Theory Ramifications, 9(8)(2000), 975-986. https://doi.org/10.1142/S0218216500000578
  5. L. H. Kauffman, Invariants of graphs in three-space, Trans. Amer. Math. Soc., 311(2)(1989), 697-710. https://doi.org/10.1090/S0002-9947-1989-0946218-0
  6. L. H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc., 318(2)(1990), 417-471. https://doi.org/10.1090/S0002-9947-1990-0958895-7
  7. L. H. Kauffman, Knots and Physics, Third edition. Series on Knots and Everything, Vol. 1, World Sci. Pub. (2001).
  8. L. H. Kauffman and P. Vogel, Link polynomials and a graphical calculus, J. Knot Theory Ramifications, 1(1992), 59-104. https://doi.org/10.1142/S0218216592000069
  9. A. Ishii, Moves and invariants for knotted handlebodies, Algebr. Geom. Topol., 8(2008), 1403-1418. https://doi.org/10.2140/agt.2008.8.1403
  10. J. Murakami, The Kauffman polynomial of links and representation theory, Osaka J. Math., 24(1987), 745-758.
  11. H. Murakami, T. Ohtsuki and S. Yamada, Homfly polynomial via an invariant of colored plane graphs, L'Enseignement Mathematique, 44(1998), 325-360.
  12. S. Suzuki, On linear graphs in 3-sphere, Osaka J. Math., 7(1970), 375-396.