DOI QR코드

DOI QR Code

On the Braid Index of Kanenobu Knots

Takioka, Hideo

  • Received : 2013.11.08
  • Accepted : 2014.07.14
  • Published : 2015.03.23

Abstract

We study the braid indices of the Kanenobu knots. It is known that the Kanenobu knots have the same HOMFLYPT polynomial and the same Khovanov-Rozansky homology. The MFW inequality is known for giving a lower bound of the braid index of a link by applying the HOMFLYPT polynomial. Therefore, it is not easy to determine the braid indices of the Kanenobu knots. In our previous paper, we gave upper bounds and sharper lower bounds of the braid indices of the Kanenobu knots by applying the 2-cable version of the zeroth coefficient HOMFLYPT polynomial. In this paper, we give sharper upper bounds of the braid indices of the Kanenobu knots.

Keywords

braid index;HOMFLYPT polynomial;Kanenobu knot;2-cable knot

References

  1. J. W. Alexander, A lemma on systems of knotted curves, Proc. Nat. Acad. Sci. U. S. A., 9(1923), 93-95. https://doi.org/10.1073/pnas.9.3.93
  2. J. Franks and R. F. Williams, Braids and the Jones Polynomial, Trans. Amer. Math. Soc., 303(1987), 97-108. https://doi.org/10.1090/S0002-9947-1987-0896009-2
  3. P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millett, and A. Ocneanu, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc., 12(1985), 239-246. https://doi.org/10.1090/S0273-0979-1985-15361-3
  4. T. Kanenobu, Infinitely many knots with the same polynomial invariant, Proc. Amer. Math. Soc., 97(1986), 158-162. https://doi.org/10.1090/S0002-9939-1986-0831406-7
  5. T. Kanenobu, A skein relation for the HOMFLYPT polynomials of two-cable links, Algebr. Geom. Topol., 7(2007), 1211-1232. https://doi.org/10.2140/agt.2007.7.1211
  6. K. Kodama, http://www.math.kobe-u.ac.jp/HOME/kodama/knot.html
  7. W. B. R. Lickorish and K. Millett, A polynomial invariant of oriented links, Topology, 26(1987), 107-141.
  8. A. Lobb, The Kanenobu knots and Khovanov-Rozansky homology, Proc. Amer. Math. Soc., 142(2014), 1447-1455. https://doi.org/10.1090/S0002-9939-2014-11863-6
  9. H. R. Morton, Seifert circles and knot polynomials, Math. Proc. Cambridge Philos. Soc., 99(1986), 107-109. https://doi.org/10.1017/S0305004100063982
  10. J. H. Przytycki and P. Traczyk, Invariants of links of Conway type, Kobe J. Math., 4(1987), 115-139.
  11. H. Takioka, The zeroth coefficient HOMFLYPT polynomial of a 2-cable knot, J. Knot Theory Ramifications, 22(2)(2013), 1350001. https://doi.org/10.1142/S0218216513500016
  12. R. F. Williams, The braid index of generalized cables, Pacific J. Math., 155(1992), 369-375. https://doi.org/10.2140/pjm.1992.155.369

Cited by

  1. On the arc index of Kanenobu knots vol.26, pp.04, 2017, https://doi.org/10.1142/S0218216517500158
  2. Infinitely many knots with the trivial (2,1)-cable Γ-polynomial vol.27, pp.02, 2018, https://doi.org/10.1142/S021821651850013X
  3. The (2,1)-cable Γ-polynomials of knots up to ten crossings vol.27, pp.04, 2018, https://doi.org/10.1142/S0218216518500281