## Tokyo Journal of Mathematics

### The Number of Vogel Operations to Deform a Link Diagram to a Closed Braid

#### Abstract

Vogel showed that any oriented link diagram $D$ can be deformed to a closed braid by a finite sequence of Reidemeister II moves, each performed on two coherently oriented edges in a face of $D$ such that the edges are contained in distinct Seifert circles. We show that the number of such moves is constant for a given oriented link diagram, and does not depend on the sequence of moves. An easy way of calculating the number is given.

#### Article information

Source
Tokyo J. Math., Volume 28, Number 2 (2005), 299-307.

Dates
First available in Project Euclid: 5 June 2009

https://projecteuclid.org/euclid.tjm/1244208192

Digital Object Identifier
doi:10.3836/tjm/1244208192

Mathematical Reviews number (MathSciNet)
MR2191051

Zentralblatt MATH identifier
1090.57006

#### Citation

HAYASHI, Chuichiro; SAEKI, Hiroko. The Number of Vogel Operations to Deform a Link Diagram to a Closed Braid. Tokyo J. Math. 28 (2005), no. 2, 299--307. doi:10.3836/tjm/1244208192. https://projecteuclid.org/euclid.tjm/1244208192

#### References

• J. W. Alexander, A lemma on a system of knotted curves, Proc. Nat. Acad. Sci. USA. 9 (1923), 93–95.
• P. Vogel, Representation of links by braids: A new algorithm, Comment. Math. Helvetici 65 (1990), 104–113.
• S. Yamada, The minimal number of Seifert circles equals the braid index of a link, Invent. Math. 89 (1987), 347–356.