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

 

Minimal sequences of Reidemeister moves on diagrams of torus knots


Authors: Chuichiro Hayashi and Miwa Hayashi
Journal: Proc. Amer. Math. Soc. 139 (2011), 2605-2614
MSC (2010): Primary 57M25
Published electronically: December 23, 2010
MathSciNet review: 2784830
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Abstract: Let $ D(p,q)$ be the usual knot diagram of the $ (p,q)$-torus knot; that is, $ D(p,q)$ is the closure of the $ p$-braid $ (\sigma_1^{-1} \sigma_2^{-1} \cdots \sigma_{p-1}^{-1})^q$. As is well-known, $ D(p,q)$ and $ D(q,p)$ represent the same knot. It is shown that $ D(n+1,n)$ can be deformed to $ D(n,n+1)$ by a sequence of $ \{ (n-1)n(2n-1)/6 \} + 1$ Reidemeister moves, which consists of a single RI move and $ (n-1)n(2n-1)/6$ RIII moves. Using cowrithe, we show that this sequence is minimal over all sequences which bring $ D(n+1,n)$ to $ D(n,n+1)$.


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Additional Information

Chuichiro Hayashi
Affiliation: Department of Mathematical and Physical Sciences, Faculty of Science, Japan Women’s University, 2-8-1 Mejirodai, Bunkyo-ku, Tokyo, 112-8681, Japan
Email: hayashic@fc.jwu.ac.jp

Miwa Hayashi
Affiliation: Department of Mathematical and Physical Sciences, Faculty of Science, Japan Women’s University, 2-8-1 Mejirodai, Bunkyo-ku, Tokyo, 112-8681, Japan
Email: miwakura@fc.jwu.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10800-6
PII: S 0002-9939(2010)10800-6
Keywords: Knot diagram, Reidemeister move, cowrithe, torus knot, positive knot.
Received by editor(s): March 10, 2010
Received by editor(s) in revised form: June 19, 2010
Published electronically: December 23, 2010
Additional Notes: The first author is partially supported by Grant-in-Aid for Scientific Research (No. 18540100), Ministry of Education, Science, Sports and Technology, Japan
Dedicated: Dedicated to Professor Akio Kawauchi for his 60th birthday
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.