Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

ON BRAID-PLAT RELATIONS IN CONWAY FUNCTION

  • Yun, Ki-Heon (Department of Mathematics, Sungshin Women's University)
  • Received : 2011.08.02
  • Accepted : 2011.08.21
  • Published : 2011.09.25
Download PDF
⟨ Previous Next ⟩

Abstract

There are two kinds of closing method for a given braid ${\beta}{\in}B_{2n}$, a braid closure $\hat{\beta}$ and a plat closure $\bar{\beta}$. In the article, we find a relation between the Conway potential function ${\nabla}_{\hat{\beta}}$ of braid closure $\hat{\beta}$ and ${\nabla}_{\hat{\beta}}$ of plat closure $\bar{\beta}$.

Keywords

References

  1. J. S. Birman, Braids, links, and mapping class groups, Princeton University Press, Princeton, N.J., 1974, Annals of Mathematics Studies, No. 82.
  2. J. S. Birman and Taizo Kanenobu, Jones' braid-plat formula and a new surgery triple, Proc. Amer. Math. Soc. 102 (1988), no. 3, 687-695. MR89c:57003
  3. R. Fintushel and R. Stern, Knots, links, and 4-manifolds, Inventiones Mathematicae 134 (1998), 363-400. https://doi.org/10.1007/s002220050268
  4. J. A. Hillman, The Torres conditions are insufficient, Math. Proc. Cambridge Philos. Soc. 89 (1981), no. 1, 19-22. https://doi.org/10.1017/S0305004100057893
  5. J. Hoste, Sewn-up r-link exteriors, Pacific J. Math. 112 (1984), no. 2, 347-382. https://doi.org/10.2140/pjm.1984.112.347
  6. M. E. Kidwell, Alexander polynomials of links of small order, Illinois J. Math. 22 (1978), no. 3, 459-475.
  7. J. Levine, A characterization of knot polynomials, Topology 4 (1965), 135-141. https://doi.org/10.1016/0040-9383(65)90061-3
  8. J. Levine, A method for generating link polynomials, Amer. J. Math. 89 (1967), 69-84. https://doi.org/10.2307/2373097
  9. J. W. Morgan, Tomasz S. Mrowka, and Zoltan Szabo, Product formulas along $T^3$ for Seiberg-Witten invariants, Math. Res. Lett. 4 (1997), no. 6, 915-929. https://doi.org/10.4310/MRL.1997.v4.n6.a11
  10. H. R. Morton, Fibred knots with a given Alexander polynomial, Knots, braids and singularities (Plans-sur-Bex, 1982), Monogr. Enseign. Math., vol. 31, Enseignement Math., Geneva, 1983, pp. 205-222.
  11. H. R. Morton, The multivariable Alexander polynomial for a closed braid, Low Dimensional Topology, Contemporary Mathematics, vol. 233, American mathematical Society, 1998, pp. 167-172.
  12. J. Murakami, A state model for the multivariable Alexander polynomial, Pacific J. Math. 157 (1993), no. 1, 109-135. https://doi.org/10.2140/pjm.1993.157.109
  13. H. Seifert, Uber das Geschlecht von Knoten, Math. Ann. 110(1935), no. 1, 571-592. https://doi.org/10.1007/BF01448044
  14. G. Torres, On the Alexander polynomial, Ann. of Math. (2) 57 (1953), 57-89. https://doi.org/10.2307/1969726
  15. V.G Turaev, Reidermeister torsion in knot theory, Russian math. Surveys 41 (1986), no. 1, 119-182. https://doi.org/10.1070/RM1986v041n01ABEH003204
  16. V.G Turaev, Introduction to combinatorial torsions, Lectures in Mathematics ETH Zurich, Birkhauser Verlag, Basel, 2001, Notes taken by Felix Schlenk.