[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/JKMS.2015.52.2.389

THE JONES POLYNOMIAL OF KNOTS WITH SYMMETRIC UNION PRESENTATIONS

Tanaka, Toshifumi (Department of Mathematics Faculty of Education Gifu University)

Publication Information

Journal of the Korean Mathematical Society / v.52, no.2, 2015 , pp. 389-402 More about this Journal

Abstract

A symmetric union is a diagram of a knot, obtained from diagrams of a knot in the 3-space and its mirror image, which are symmetric with respect to an axis in the 2-plane, by connecting them with 2-tangles with twists along the axis and 2-tangles with no twists. This paper presents an invariant of knots with symmetric union presentations, which is called the minimal twisting number, and the minimal twisting number of $10_{42}$ is shown to be two. This paper also presents a sufficient condition for non-amphicheirality of a knot with a certain symmetric union presentation.

Keywords

symmetric union; Jones polynomial; ribbon knot;

Citations & Related Records

Reference

1	M. Eisermann, The Jones polynomial of ribbon links, Geom. Topol. 13 (2009), no. 2, 623-660. DOI
2	M. Eisermann and C. Lamm, Equivalence of symmetric union diagrams, J. Knot Theory Ramifications 16 (2007), no. 7, 879-898. DOI ScienceOn
3	J. A. Hillman, Alexander Ideals of Links, Lecture Notes in Mathematics, Vol. 895. Springer-Verlag, Berlin-New York, 1981.
4	V. F. R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 1, 103-111. DOI
5	L. H. Kauffman, State models and the Jones polynomials, Topology 26 (1987), no. 3, 395-407. DOI ScienceOn
6	S. Kinoshita and H. Terasaka, On unions of knots, Osaka J. Math. 9 (1957), 131-153.
7	C. Lamm, Symmetric unions and ribbon knots, Osaka J. Math. 37 (2000), no. 3, 537- 550.
8	C. Lamm, Symmetric unions and ribbon knots, Symmetric union presentations for 2-bridge ribbon knots, arXiv:math.GT/ 0602395, 2006.
9	W. B. R. Lickorish, An Introduction to Knot Theory, Graduate Texts in Mathematics, Vol. 175, Springer-Verlag, New York, 1997.
10	W. B. R. Lickorish and K. C. Millett, Some evaluations of link polynomials, Comment. Math. Helv. 61 (1986), no. 3, 349-359. DOI
11	P. Lisca, Lens spaces, rational balls and the ribbon conjecture, Geom. Topol. 11 (2007), 429-472. DOI
12	H. Murakami, A recursive calculation of the Arf invariant of a link, J. Math. Soc. Japan 38 (1986), no. 2, 335-338. DOI
13	K. Murasugi, Jones polynomials and classical conjectures in knot theory, Topology 26 (1987), no. 2, 187-194. DOI ScienceOn
14	L. Watson, Knots with identical Khovanov homology, Algebr. Geom. Topol. 7 (2007), 1389-1407. DOI