JOURNAL BROWSE
Search
Advanced SearchSearch Tips
On the Braid Index of Kanenobu Knots
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
  • Journal title : Kyungpook mathematical journal
  • Volume 55, Issue 1,  2015, pp.169-180
  • Publisher : Department of Mathematics, Kyungpook National University
  • DOI : 10.5666/KMJ.2015.55.1.169
 Title & Authors
On the Braid Index of Kanenobu Knots
Takioka, Hideo;
  PDF(new window)
 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;
 Language
English
 Cited by
 References
1.
J. W. Alexander, A lemma on systems of knotted curves, Proc. Nat. Acad. Sci. U. S. A., 9(1923), 93-95. crossref(new window)

2.
J. Franks and R. F. Williams, Braids and the Jones Polynomial, Trans. Amer. Math. Soc., 303(1987), 97-108. crossref(new window)

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. crossref(new window)

4.
T. Kanenobu, Infinitely many knots with the same polynomial invariant, Proc. Amer. Math. Soc., 97(1986), 158-162. crossref(new window)

5.
T. Kanenobu, A skein relation for the HOMFLYPT polynomials of two-cable links, Algebr. Geom. Topol., 7(2007), 1211-1232. crossref(new window)

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. crossref(new window)

9.
H. R. Morton, Seifert circles and knot polynomials, Math. Proc. Cambridge Philos. Soc., 99(1986), 107-109. crossref(new window)

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. crossref(new window)

12.
R. F. Williams, The braid index of generalized cables, Pacific J. Math., 155(1992), 369-375. crossref(new window)