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 be the usual knot diagram of the -torus knot; that is, is the closure of the -braid . As is well-known, and represent the same knot. It is shown that can be deformed to by a sequence of Reidemeister moves, which consists of a single RI move and RIII moves. Using cowrithe, we show that this sequence is minimal over all sequences which bring to .

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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:
https://doi.org/10.1090/S0002-9939-2010-10800-6

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.