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Stick numbers of $ 2$-bridge knots and links


Authors: Youngsik Huh, Sungjong No and Seungsang Oh
Journal: Proc. Amer. Math. Soc. 139 (2011), 4143-4152
MSC (2010): Primary 57M25, 57M27
DOI: https://doi.org/10.1090/S0002-9939-2011-10832-3
Published electronically: March 16, 2011
MathSciNet review: 2823059
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Abstract: Negami found an upper bound on the stick number $ s(K)$ of a nontrivial knot $ K$ in terms of the minimal crossing number $ c(K)$ of the knot, which is $ s(K) \leq 2 c(K)$. Furthermore, McCabe proved that $ s(K) \leq c(K) + 3$ for a $ 2$-bridge knot or link, except in the cases of the unlink and the Hopf link. In this paper we construct any $ 2$-bridge knot or link $ K$ of at least six crossings by using only $ c(K)+2$ straight sticks. This gives a new upper bound on stick numbers of $ 2$-bridge knots and links in terms of crossing numbers.


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  • [ABGW] Colin C. Adams, Bevin M. Brennan, Deborah L. Greilsheimer and Alexander K. Woo, Stick numbers and composition of knots and links, J. Knot Theory Ramif. 6 (1997) 149-161. MR 1452436 (98h:57010)
  • [BP] Y. Bae and C. Park, An upper bound of arc index of links, Math. Proc. Camb. Phil. Soc. 129 (2000) 491-500. MR 1780500 (2002f:57009)
  • [BZ] G. Burde and H. Zieschang, Knots, Walter de Gruyter & Co., Berlin (1985). MR 808776 (87b:57004)
  • [Ca] J. Calvo, Characterizing polygons in $ \mathbb{R}^3$, in Physical Knots, Contemporary Mathematics, 304, Amer. Math. Soc. (2002) 37-53. MR 1953010 (2003m:57006)
  • [Co] J. Conway, An enumeration of knots and links, and some of their algebraic properties, in Computational Problems in Abstract Algebra, Pergamon Press, New York (1970) 329-358. MR 0258014 (41:2661)
  • [FLS] E. Furstenberg, J. Li and J. Schneider, Stick knots, Chaos, Solitons & Fractals 9 (1998) 561-568. MR 1628742 (99g:57007)
  • [HO] Y. Huh and S. Oh, An upper bound on stick numbers of knots, to appear in J. Knot Theory Ramif.
  • [J] G. T. Jin, Polygon indices and superbridge indices of torus knots and links, J. Knot Theory Ramif. 6 (1997) 281-289. MR 1452441 (98h:57018)
  • [K] L. Kauffman, State models and the Jones polynomial, Topology 26 (1987) 395-407. MR 899057 (88f:57006)
  • [Mc] C. L. McCabe, An upper bound on edge numbers of $ 2$-bridge knots and links, J. Knot Theory Ramif. 7 (1998) 797-805. MR 1643867 (99h:57015)
  • [Me] M. Meissen, homepage at http://www.bethelks.edu/meissen.
  • [Mu] K. Murasugi, Jones polynomials and classical conjectures in knot theory, Topology 26 (1987) 187-194. MR 895570 (88m:57010)
  • [N] S. Negami, Ramsey theorems for knots, links, and spatial graphs, Trans. Amer. Math. Soc. 324 (1991) 527-541. MR 1069741 (92h:57014)
  • [R] R. Randell, An elementary invariant of knots, J. Knot Theory Ramif. 3 (1994) 279-286. MR 1291860 (95e:57019)
  • [T] M. Thistlethwaite, A spanning tree expansion of the Jones polynomial, Topology 26 (1987) 297-309. MR 899051 (88h:57007)

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Additional Information

Youngsik Huh
Affiliation: Department of Mathematics, School of Natural Sciences, Hanyang University, Seoul 133-791, Republic of Korea
Email: yshuh@hanyang.ac.kr

Sungjong No
Affiliation: Department of Mathematics, Korea University, 1, Anam-dong, Sungbuk-ku, Seoul 136-701, Republic of Korea
Email: blueface@korea.ac.kr

Seungsang Oh
Affiliation: Department of Mathematics, Korea University, 1, Anam-dong, Sungbuk-ku, Seoul 136-701, Republic of Korea
Email: seungsang@korea.ac.kr

DOI: https://doi.org/10.1090/S0002-9939-2011-10832-3
Keywords: Knot, stick number, 2-bridge
Received by editor(s): March 9, 2010
Received by editor(s) in revised form: July 22, 2010, and September 16, 2010
Published electronically: March 16, 2011
Additional Notes: This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MEST) (No. 2009-0074101).
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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