A new proof of Markov's braid theorem

Paweł Traczyk

Banach Center Publications (1998)

  • Volume: 42, Issue: 1, page 409-419
  • ISSN: 0137-6934

Abstract

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The purpose of this paper is to introduce a new proof of Markov's braid theorem, in terms of Seifert circles and Reidemeister moves. This means that the proof will be of combinatorial and essentially 2-dimensional nature. One characteristic feature of our approach is that nowhere in the proof will we use or refer to the braid axis. This allows for greater flexibility in various transformations of the diagrams considered. Other proofs of Markov's theorem can be found in [2], [3], [4] and [5].

How to cite

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Traczyk, Paweł. "A new proof of Markov's braid theorem." Banach Center Publications 42.1 (1998): 409-419. <http://eudml.org/doc/208820>.

@article{Traczyk1998,
abstract = {The purpose of this paper is to introduce a new proof of Markov's braid theorem, in terms of Seifert circles and Reidemeister moves. This means that the proof will be of combinatorial and essentially 2-dimensional nature. One characteristic feature of our approach is that nowhere in the proof will we use or refer to the braid axis. This allows for greater flexibility in various transformations of the diagrams considered. Other proofs of Markov's theorem can be found in [2], [3], [4] and [5].},
author = {Traczyk, Paweł},
journal = {Banach Center Publications},
keywords = {Markov moves; braid},
language = {eng},
number = {1},
pages = {409-419},
title = {A new proof of Markov's braid theorem},
url = {http://eudml.org/doc/208820},
volume = {42},
year = {1998},
}

TY - JOUR
AU - Traczyk, Paweł
TI - A new proof of Markov's braid theorem
JO - Banach Center Publications
PY - 1998
VL - 42
IS - 1
SP - 409
EP - 419
AB - The purpose of this paper is to introduce a new proof of Markov's braid theorem, in terms of Seifert circles and Reidemeister moves. This means that the proof will be of combinatorial and essentially 2-dimensional nature. One characteristic feature of our approach is that nowhere in the proof will we use or refer to the braid axis. This allows for greater flexibility in various transformations of the diagrams considered. Other proofs of Markov's theorem can be found in [2], [3], [4] and [5].
LA - eng
KW - Markov moves; braid
UR - http://eudml.org/doc/208820
ER -

References

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  1. [1] J.W. Alexander, A Lemma on Systems of Knotted Curves, Proc. Nat. Acad. Sci. 9 (1923), 93-95. 
  2. [2] D. Bennequin, Entrelacements et équations de Pfaff, Astérisque 107-108 (1983), 87-161. 
  3. [3] J. Birman, Braids, links and the mapping class groups, Annals of Math. Stud. 82, Princeton University Press, 1974. 
  4. [4] A.A. Markov, Über die freie Äquivalenz geschlossener Zöpfe, Recueil Mathématique Moscou 1 (1935). 
  5. [5] H.R. Morton, Threading knot diagrams, Math. Proc. Camb. Phil. Soc. 99 (1986), 247-260. Zbl0595.57007
  6. [6] P. Vogel, Representation of links by braids: A new algorithm, Comment. Math. Helvetici 65 (1990), 104-113. Zbl0703.57004
  7. [7] S. Yamada, The minimal number of Seifert circles equals the braid index of a link, Invent. Math. 89 (1987), 347-356. Zbl0634.57004

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