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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Stick numbers of $2$-bridge knots and links
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by Youngsik Huh, Sungjong No and Seungsang Oh PDF
Proc. Amer. Math. Soc. 139 (2011), 4143-4152 Request permission

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
  • 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
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2823059