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].
@article{bwmeta1.element.bwnjournal-article-bcpv42i1p409bwm,
author = {Traczyk, Pawe\l },
title = {A new proof of Markov's braid theorem},
journal = {Banach Center Publications},
volume = {43},
year = {1998},
pages = {409-419},
zbl = {0901.57018},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv42i1p409bwm}
}
Traczyk, Paweł. A new proof of Markov's braid theorem. Banach Center Publications, Tome 43 (1998) pp. 409-419. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv42i1p409bwm/
[000] [1] J.W. Alexander, A Lemma on Systems of Knotted Curves, Proc. Nat. Acad. Sci. 9 (1923), 93-95.
[001] [2] D. Bennequin, Entrelacements et équations de Pfaff, Astérisque 107-108 (1983), 87-161.
[002] [3] J. Birman, Braids, links and the mapping class groups, Annals of Math. Stud. 82, Princeton University Press, 1974.
[003] [4] A.A. Markov, Über die freie Äquivalenz geschlossener Zöpfe, Recueil Mathématique Moscou 1 (1935).
[004] [5] H.R. Morton, Threading knot diagrams, Math. Proc. Camb. Phil. Soc. 99 (1986), 247-260. | Zbl 0595.57007
[005] [6] P. Vogel, Representation of links by braids: A new algorithm, Comment. Math. Helvetici 65 (1990), 104-113. | Zbl 0703.57004
[006] [7] S. Yamada, The minimal number of Seifert circles equals the braid index of a link, Invent. Math. 89 (1987), 347-356. | Zbl 0634.57004