Abstract
The terms “braid” and “braid groups” were coined by Artin, 1925. In his paper, an n-braid appears as a specific topological object. We consider two parallel planes in euclidean 3-space which we call respectively the upper and the lower frame. We choose n distinct points U v (v = 1, ..., n) in the upper frame and denote their orthogonal projections onto the lower frame by L v . Next, we join each U v with an L μ by a polygon which intersects any plane between (and parallel to) the upper and lower frame exactly once. These polygons are called strings. We assume that they do not intersect anywhere, and that v → μ(v) is a permutation of the symbols 1, ..., n. By removing the strings from the slice of 3-space between upper and lower frame, we obtain an open subset of 3-space the isotopy class of which we call an n-braid. We define a composition between n-braids by hanging on one n braid to the other one. (This can be done by identifying the upper frame of the second braid with the lower frame of the first one, removing these two frames and compressing the slice of 3-space between the first upper and the second lower frame by an affine transformation to the same thickness as before.) Under this composition, the n-braids form a group B n which has n − 1 generators σ v . These are represented respectively by braids which have a projection onto a plane perpendicular to the frames in which the v-th and (v+l)-st string seem to cross once whereas all other strings go straight through as line segments orthogonal to both frames. The braid represented by n strings which go straight through is the representative of the unit element of B n . (Figure 1 shows a representative of the particular 3-braid σ1σ −12 σ1σ −12 .) Of course, this rather vague definition of B n can be made rigorous. See Artin 1947a, Burde 1963.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
J.W. Alexander, II, (1923), “A lemma on systems of knotted curves”, Proc. Nat. Acad. USA 9, 93–95.
B.N. Арнольд [V.I. Arnol’d], (1968), “Замечние о ветвлении гиперэлиптичесних интегралов кан функций параметров” [A remark on the branching of hyperelliptic integrals as functions of the parameters], Funkcional. Anal. i Puloćen. 2, no. 3, 1–3; Functicnal Anal. Appi. 2, 187–189.
B.И. Арнолъд [V.I. Arnol’d], (1968), “О носах алгебраических функций и когомологиях ласточкиных хвоснов” [Fibrations of algebraic functions and cohomologies of dovetails], Uspehi Mat. Nauk 23, no. 4, 247–248.
B.И. Арнолъд [V.I. Arnol’d], (1969), “Колъцо когомологий группы грашеных нос” [The cohomology ring of the group of dyed braids], Mut. Zametki 5, 227–231.
B.И. Арнолъд [V.I. Arnol’d], (1970), “Топологические инварианты алгебраических функций ІІ” [Topological invariants of algebraic functions, II], Functional. Anal. i Prilozen. 4, no. 2, 1–9; Functional Anal. Appi. 4, 91–98.
Emil Artin, (1925), “Theorie der Zöpfe”, Abri. Math. Seri. Univ. Horhura 4 (1926), 47–72.
E. Artin, (1947a), “Theory of braids”, Ann. of Math. (2) 48, 101–126.
E. Artin, (1947b), “Braids and permutations”, Ann. of Math. (2) 48, 643–649.
S. Bachmuth, (1965), “Automorphisms of free metabelian groups”, Trans. Amer. Math. Soc. 118, 93–104.
C.T. Benson and L.C. Grove, (1971), Finite reflection groups ( Bogden and Quigley, Tarrytown on Hudson New York).
P. Bergau und J. Mennicke, (1960), “Über topologische Abbildungen der Brezelfläche vom Geschlecht 2”, Math. Z. 74, 414–435.
Joan S. Birman, (1969a), “On braid groups”, Comm. Pure Appl. Math. 22, 41–72.
Joan S. Birman, (1969h), “Mapping class groups and their relationship to braid groups”, Comm. Pure Appl. Math. 22, 213–238.
Joan S. Birman, (1969c), “Non-conjugate braids can define isotopic knots”, Comm. Pure Appl. Math. 22, 239–242.
Joan S. Birman, (1973), “Plat representations for link groups”, Comm. Pure Appi. Math. 26
Joan S. Birman, (to appear), “Braids, links and mapping class groups”, A research ronograph (Annals of Mathematics Studies).
Jcan S. Birman, (to appear), “Mapping class groups of surfaces: a survey”, Proc. Third Conf, on Riemann Surfaces, 1974 ( Annals of Mathematics Studies).
Joan S. Birman and Hugh M. Hilden, (1971), “On the mapping class groups of closed surfaces as covering spaces”, Advances in the theory of Riemann surfaces, pp. 81–115 (Proc. 1969 Stony Brook Conf. Annals of Mathematics Studies, 66. Princeton University Press and University of Tokyo Press, Princeton New Jersey). MR45#1169.
Joan S. Birman and Hugh M. Hilden, (1972), “Isotopies of homcomorphisms of Riemann surfaces and a theorem about Artin’s braid group”, Bull. Amer. Math. Soc. 78, 1002–1004.
F. Bohnenblust, (1947), “The algebraical braid group”, Ann. of Math. (2) 48, 127–136.
E. Brieskorn, (1971), “Pie Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe”, Invent. Math. 12, 57–61.
Egbert Brieskorn, (1973), “Sur les groupes de tresses d’aprês V.I. Arnol’d”, Seminaire Bourbaki, Vol. 1971/72, Exposé 401 (Lecture Notes in Mathematics, 317. Springer-Verlag, Berlin, Heidelberg, New York ).
Egbert Brieskorn und Kyoji Saito, (1972), “Artin-Gruppen und Coxeter-Gruppen”, Invent. Math. 17, 245–271.
Werner Burau, (1936), “Über Verkettungsgruppen”,.4bh. Math. Sem. Hansichen Univ. 11, 171–178.
Gerhard Burde, (1963), “Zur Theorie der Zöpfe”, Math. Ann. 151, 101–107.
Wei-Liang Chow, (1948), “On the algebraical braid group”, Ann. of Math. (2) 49, 654–658.
Daniel I.A. Cohen, (1967), “On representations of the braid group”, J. Algebra 7, 145–151.
David B. Cohen, (1973), “The Hurwitz monodromy group”, (PhD thesis, New York University New York).
H.S.M. Coxeter, W.O.J. Moser, (1957), Generators and relations for discrete groups (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 14. Springer-Verlag, Berlin, Göttingen, Heidelberg). MR19, 527.
Richard H. Crowell and Ralph H. Fox, (1963), Introduction to knot theory ( Ginn and Co., Boston).
Pierre Deligne, (1972), “Les immeubles des groupes de tresses généralisés”, Invent. Math. 17, 273–302.
E. Fadell, (1962), “Homotopy groups of configuration spaces and the string problem of Dirac”, Duke Math. J. 29, 231–242
Edward Fadell and Lee Neuwirth, (1952), “Configuration spaces”, Math. Scan.d. 10, 111–118.
Edward Fadell and James Van Buskirk, (1962), “The braid groups of E and Duke Math. J. 29, 243–257.
R. Fox and L. Neuwirth, (1962), “The braid groups”, Math. Stand. 10, 119–126.
R. Fricke and F. Klein, (1965), Vorlesungen über die Theorie der automorphen Funetionen (Band I. Die Gruppentheoretischen Grundlagen. Teubner, Leipzig, 1897; FdM28,334. Reprinted Johnson Reprint Corp., New York; B.G. Teubner Verlagsgesellschaft, Stuttgart). MR32#1348.
W. Fröhlich, (1936), “Ober ein spezielles Transformationsproblem bei einer besonderen Klasse von Zöpfen”, Md. Math. Phys. 44, 225–237.
F.A. Garside, (1969), “The braid group and other groups”, Quart. J. Math. Oxford Ser. (2) 20, 235–254.
Betty Jane Gassner, (1961), “On braid groups”, Ath. Math. Sem. Univ. Hamburg 25, 10–22.
Richard Gillette and James Van Buskirk, (1968), “The word problem and consequences for the braid groups and mapping class groups of the 2-sphere”, Trans. Amer. Math. Soc. 131, 277–296.
E.A. Горин, В.Я. Лин [E.A. Gorin, V.Ja. Lin], (1969), “Algebraic equations with continuous coefficients mapping class groups of the 2-sphere” [Algebraic equations with continuous coefficients, and certain questions of the algebraic theory of braids], Mat. Sb. (MS) 78 (120), 579–610; Math. USSR-Sb. 7, 569–596. MR40#4939.
A. Hurwitz, (1891), “Lieber Riemann’sche Flächen mit gegebenen Verzweigungspunkten”, Math. Ann. 39, 1–61.
A. Hurwitz, (1902), “Ueber die Anzahl der Riemann’schen Flächen mit gegebenen Verzweigungspunkten”, Math. Ann. 55, 53–66.
S. Kinoshita and H. Terasaka, (1957), “On unions of knots”, Osaka Math. J. 9, 131–153.
H. Levinson, (1973), “Decomposable braids and linkages”, Trans. Amer. Math. Soc. 178, 111–126.
Seymour Lipschutz, (1961), “On a finite matrix representation of the braid group”, Arch. Math. 12, 7–12.
Seymour Lipschutz, (1963), “Note on a paper by Shepperd on the braid group”, Proe. Amer. Math. Soc. 14, 225–227.
Colin Maclachlan, (1973), “On a Conjecture of Magnus on the Hurwitz Monodromy Group”, Math. Z. 132, 45–50.
Wilhelm Magnus, (1934), “Über Automorphismen von Fundamentalgruppen berandeter Flächen”, Math. Ann. 109, 617–646.
W. Magnus, (1972), “Braids and Riemann surfaces”, Comm. Pure Appl. Math. 25, 151–161.
Wilhelm Magnus and Ada Peluso, (1967), “On knot groups”, Comm. Pure Appl. Math. 20, 749–770.
Wilhelm Magnus and Ada Peluso, (1969), “On a theorem of V.I. Arnol’d”, Comm. Pure Appl. Math. 22, 683–692.
Г.С. Маканин [G.S. Makanin], (1968), “Проблема сопряженности в группе кос” [The conjugacy problem in the braid group], Mold. Akad. Nauk SSSR 182, 495–496; Soviet Math. Maki. 9, 1156–1157.
A. Markoff, (1936), “Über die freie Äquivalenz der geschlossenen Zöpfe”, Mat. Sb. (NS) 1 (43), 73–78.
А.А. Марков [A. Markoff], (1945), Основы алгеба ической теории кос [Foundations of the algebraic theory of tresses] (Trudy Mat. Inst. Steklov 16). MR8,131.
K. Murasugi, (1973), “On closed 3-braids”, (Preprint).
K. Murasugi and R.S.D. Thomas, (1972), “Isotopic closed nonconjugate braids”, Proc. Amer. Math. Soc. 33, 137–139.
M.H.A. Newman, (1942), “On a string problem of Dirac”, J. dondon Math. Soc. 17, 173–177.
C.D. Papakyriakopoulos, (1957), “On Dehn’s lemma and the asphericity of knots”, Ann. of Math. (2) 66, 1–26.
Kurt Reidemeister, (1960), “Knoten und Geflechte”, Nachr. Akad. Wies. Gottingen Math.-Phys. 57. 2, 105–115.
G.P. Scott, (1970), “Braid groups and the group of homeomorphisms of a surface”, Proc. Cambridge Philos. Soc. 68, 605–617.
J.A.H. Sheppard, (1962), “Braids which can be plaited with their ends tied together at each end”, Proc. Roy. Soc. London Ser. A 265, 229–244.
Jacques Tits, (1969), “Le problème des mots dans les groupes de Coxeter”, Symposia Mathematica, IN5AM, Rome, 1967/68, 1, 175–185 ( Academic Press, London, New York ).
James Van Buskirk, (1966), “Braid groups of compact 2-manifolds with elements of finite order”, Trans. Amer. Math. Soc. 122, 81–97.
О.Я. Виро [O.Ja. Viro], (1972), “Зацепления, двулистные разветвленные накрытия и косы” [Linkings, 2-sheeted branched coverings and braids], Mat. Sb. (NS) 87 (129), 216–228
О.Я. Виро: Math. USSR-Sb. 16 (1972), 223–236.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1974 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Magnus, W. (1974). Braid Groups: A Survey. In: Newman, M.F. (eds) Proceedings of the Second International Conference on the Theory of Groups. Lecture Notes in Mathematics, vol 372. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21571-5_49
Download citation
DOI: https://doi.org/10.1007/978-3-662-21571-5_49
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-06845-7
Online ISBN: 978-3-662-21571-5
eBook Packages: Springer Book Archive
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.