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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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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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: http://dx.doi.org/10.1090/S0002-9939-2011-10832-3
PII: S 0002-9939(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.