Tokyo Journal of Mathematics

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

Chuichiro HAYASHI and Hiroko SAEKI

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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

Permanent link to this document
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


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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.