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Annals of Mathematics

Theory of Braids
Author(s): E. Artin
Reviewed work(s):
Source: The Annals of Mathematics, Second Series, Vol. 48, No. 1 (Jan., 1947), pp. 101-126
Published by: Annals of Mathematics
Stable URL: http://www.jstor.org/stable/1969218 .
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ANNALS OF MATHEMATICS
Vol. 48, No. 1, January, 1947

THEORY OF BRAIDS
BY E.

ARTIN

(Received May 20, 1946)

A theoryof braids leading to a classification as givenin my paper "Theorie
w
der Zopfe" in vol. 4 of theHamburgerAbhandlungen(quoted as Z). Most of
the proofsare entirely ntuitive. That of the main theoremin ?7 is not even
i
convincing. It is possible to correctthe proofs. The difficultieshat one ent
countersif one triesto do so come fromthe fact that projectionof the braid,
whichis an excellenttool forintuitiveinvestigations,s a very clumsyone for
i
rigorous roofs. This has lead me to abandon projectionsltogether. We shall
p
a
use the more powerfultool of braid coordinatesand obtain therebyfarther
reachingresultsof greatergenerality.
A fewwordsabout the initialdefinitions. The factthat we assume of a braid
l
stringthat it ends in a straight ine is of course unimportant. It could be reo
placed by limit assumptionsor introduction f infinite oints. The present
p
definition as selectedbecauseit makes some of the discussionseasier and may
w
be replaced any time by anotherone. I also wish to stress the fact that the
o
definition f s-isotopyis of a provisionalcharacteronly and is replaced later
(Definition3) by a general notion of isotopy.
n
More than halfofthe paper is of a geometric ature. In thispart we develop
some resultsthat may escape an intuitiveinvestigation(Theorem 7to 10).
We do not prove (as has been done in Z) that the relations(18) (19) are det
fining elationsforthe braid group. We refer he readerto a paper by F. Bohr
w
nenblust1 herea proofof thisfactand of manyof our resultsis given by purely
group theoreticalmethods.
Later the proofsbecome more algebraic. With the developed tools we are
able to give a unique normalformfor every braid2(Theorem 17,fig.4 and ret
mark following heorem 18). In Theorem 19 we determine he centerof the
T
o
w
braid groupand finally e give a characterisation f braids of braids.
a
I would like to mentionin this introduction few of the more important f
o
the unsolved problems:
i
1) Assumethat two braids can be deformednto each otherby a deformation
of the most generalnatureincluding elfintersectionf eachstring ut avoiding
o
b
s
o
intersectionf two different
strings. Are theyisotopic? One would be inclined
to doubt it. Theorem8 solves, however,a special case of this problem.
2) In Definition3, we introducea notion of isotopy that is already very
general. What conditionsmust be put on a many to many mappingso that
the result of Theorem 9 still holds?
I F. Bohnenblust, The algebraicalbraid group, Ann. Math., vol. 48, (1947), pp. 127-136.
2The freedom of the group of k-pure braids has been proved with other methods in:
b
W. FR6HLICH, Uber ein spezielle8 Transformationsproblem ei einer besonderen Klasse von
Zopfen, Monatshefte fur Math. und Physik, vol. 44 (1936), p. 225.
101

102

E. ARTIN

3) Determine all automorphisms f the braid group.
o
4) With what braids is...