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Braid groups: A survey

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Book cover Proceedings of the Second International Conference on The Theory of Groups

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References

  • J.W. Alexander, II, (1923), "A lemma on systems of knotted curves", Proc. Nat. Acad. USA 9, 93–95. FdM49,408.

    Article  Google Scholar 

  • В.И. Арнольд [V.I. Arnol'd], (1968), "Замечание о ветвлении гиперэллиптических интегралов как функций параметров" [A remark on the branching of hyperelliptic integrals as functions of the parameters], Funkcional. Anal. i Puložen. 2, no. 3, 1–3; Functional Anal. Appl. 2, 187–189. MR39#5583.

    Google Scholar 

  • В.И. Арнольд [V.I. Arnol'd], (1968). "О косах алгебраических функций и когомологиях ласточкиных хвоснов" [Fibrations of algebraic functions and cohomologies of dovetails], Uspehi Mat. Nauk 23, no. 4, 247–248. MR38#156.

    MathSciNet  Google Scholar 

  • В.И. Арнольд [V.I. Arnol'd], (1969), "Кольцо когомологий группы кращеных кос" [The cohomology ring of the group of dyed braids], Mat. Zametki 5, 227–231. MR39#3529.

    MathSciNet  Google Scholar 

  • В.И. Арнольд [V.I. Arnol'd], (1970), "Топологические инвариантыалгебр аическихфункций. II" [Topological invariants of algebraic functions, II], Funktional. Anal. i Priložen. 4, no. 2, 1–9; Functional Anal. Appl. 4, 91–98. MR43#1991.

    Google Scholar 

  • Emil Artin, (1925), "Theorie der Zöpfe", Abh. Math. Sem. Univ. Hamburg 4 (1926), 47–72. FdM51,450.

    Article  MathSciNet  MATH  Google Scholar 

  • E. Artin, (1947a), "Theory of braids", Ann. of Math. (2) 48, 101–126. FdM8,367.

    Google Scholar 

  • E. Artin, (1947b), "Braids and permutations", Ann. of Math. (2) 48, 643–649. MR9,6.

    Article  MathSciNet  MATH  Google Scholar 

  • S. Bachmuth, (1965), "Automorphisms of free metabelian groups", Trans. Amer. Math. Soc. 118, 93–104. MR31#4831.

    Article  MathSciNet  MATH  Google Scholar 

  • C.T. Benson and L.C. Grove, (1971), Finite reflection groups (Bogden and Quigley, Tarrytown on Hudson, New York).

    MATH  Google Scholar 

  • P. Bergau und J. Mennicke, (1960), "Über topologische Äbbildungen der Brezelfläche vom Geschlecht 2", Math. Z. 74, 414–435. MR27#1960.

    Article  MathSciNet  MATH  Google Scholar 

  • Joan S. Birman, (1969a), "On braid groups", Comm. Pure Appl. Math. 22, 41–72. MR38#2764.

    Article  MathSciNet  MATH  Google Scholar 

  • Joan S. Birman, (1969b), "Mapping class groups and their relationship to braid groups", Comm. Pure Appl. Math. 22, 213–238. MR39#4840.

    Article  MathSciNet  MATH  Google Scholar 

  • Joan S. Birman, (1969c), "Non-conjugate braids can define isotopic knots", Comm. Pure Appl. Math. 22, 239–242. MR39#6298.

    Article  MathSciNet  MATH  Google Scholar 

  • Joan S. Birman, (1973), "Plat representations for link groups", Comm. Pure Appl. Math. 26

    Google Scholar 

  • Joan S. Birman, (to appear), "Braids, links and mapping class groups", A research monograph (Annals of Mathematics Studies).

    Google Scholar 

  • Joan S. Birman, (to appear), "Mapping class groups of surfaces: a survey", Proc. Third Conf. on Riemann Surfaces, 1974 (Annals of Mathematics Studies).

    Google Scholar 

  • 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.

    Google Scholar 

  • Joan S. Birman and Hugh M. Hilden, (1972), "Isotopies of homeomorphisms of Riemann surfaces and a theorem about Artin's braid group", Bull. Amer. Math. Soc. 78, 1002–1004.

    Article  MathSciNet  MATH  Google Scholar 

  • F. Bohnenblust, (1947), "The algebraical braid group", Ann. of Math. (2) 48, 127–136. MR8,367.

    Article  MathSciNet  MATH  Google Scholar 

  • E. Brieskorn, (1971), "Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe", Invent. Math. 12, 57–61. MR45#2692.

    Article  MathSciNet  MATH  Google Scholar 

  • Egbert Brieskorn, (1973), "Sur les groupes de tresses d'après V.I. Arnol'd", Seminaire Bourbaki, Vol. 1971/72, Exposé 401 (Lecture Notes in Mathematics, 317. Springer-Verlag, Berlin, Heidelberg, New York).

    Google Scholar 

  • Egbert Brieskorn und Kyoji Saito, (1972) "Artin-Gruppen und Coxeter-Gruppen", Invent. Math. 17, 245–271. Zbl.243.20037.

    Article  MathSciNet  MATH  Google Scholar 

  • Werner Burau, (1936), "Über Verkettungsgruppen", Abh. Math. Sem. Hansichen Univ. 11, 171–178. FdM61,1021.

    Article  MATH  Google Scholar 

  • Gerhard Burde, (1963), "Zur Theorie der Zöpfe", Math. Ann. 151, 101–107. MR27#1942.

    Article  MathSciNet  MATH  Google Scholar 

  • Wei-Liang Chow, (1948), "On the algebraical braid group", Ann. of Math. (2) 49, 654–658. MR10,98.

    Article  MathSciNet  MATH  Google Scholar 

  • Daniel I.A. Cohen, (1967), "On representations of the braid group", J. Algebra 7, 145–151. MR38#220.

    Article  MathSciNet  MATH  Google Scholar 

  • David B. Cohen, (1973), "The Hurwitz monodromy group", (PhD thesis, New York University, New York).

    MATH  Google Scholar 

  • H.S.M. Coxeter, W.O.J. Moser, (1957), Genrators and relations for discrete groups (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 14. Springer-Verlag, Berlin, Göttingen, Heidelberg). MR19,527.

    Book  MATH  Google Scholar 

  • Richard H. Crowell and Ralph H. Fox, (1963), Introduction to knot theory (Ginn & Co., Boston). MR26#4348.

    MATH  Google Scholar 

  • Pierre Deligne, (1972), "Les immeubles des groupes de tresses généralisés", Invent. Math. 17, 273–302. Zbl.238.20034.

    Article  MathSciNet  MATH  Google Scholar 

  • E. Fadell, (1962), "Homotopy groups of configuration spaces and the string problem of Dirac", Duke Math. J. 29, 231–242. MR25#4538.

    Article  MathSciNet  MATH  Google Scholar 

  • Edward Fadell and Lee Neuwirth, (1962), "Configuration spaces", Math. Scand. 10, 111–118. MR25#4537.

    Article  MathSciNet  MATH  Google Scholar 

  • Edward Fadell and James Van Buskirk, (1962), "The braid groups of E 2 and S 2", Duke Math. J. 29, 243–257. MR25#4539.

    Article  MathSciNet  MATH  Google Scholar 

  • R. Fox and L. Neuwirth, (1962), "The braid groups", Math. Scand. 10, 119–126. MR27#742.

    Article  MathSciNet  MATH  Google Scholar 

  • R. Fricke und F. Klein, (1965), Vorlesungen über die Theorie der automorphen Functionen (Band I. Die Gruppentheoretischen Grundlagen. Teubner, Leipzig, 1897; FdM28,334. Reprinted Johnson Reprint Corp., New York; B.G. Taubner Verlagsgesellschaft, Stuttgart). MR32#1348.

    MATH  Google Scholar 

  • W. Fröhlich, (1936), "Über ein spezielles Transformationsproblem bei einer besonderen Klasse von Zöpfen", Mh. Math. Phys. 44, 225–237. FdM62,658.

    Article  MATH  Google Scholar 

  • F.A. Garside, (1969), "The braid group and other groups", Quart. J. Math. Oxford Ser. (2) 20, 235–254. MR40#2051.

    Article  MathSciNet  MATH  Google Scholar 

  • Betty Jane Gassner, (1961), "On braid groups", Abh. Math. Sem. Univ. Hamburg 25, 10–22. MR24#A174.

    Article  MathSciNet  MATH  Google Scholar 

  • 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. MR38#221.

    MathSciNet  MATH  Google Scholar 

  • Е.А. Горин, В.Я. Лин [E.A. Gorin, V.Ja. Lin], (1969), "Алгебраические уравнения с непрерывными коэффициентами и некоторые вопросы алгебраической теории кос" [Algebraic equations with continuous coefficients, and certain questions of the algebraic theory of braids], Mat. Sb. (NS) 78 (120), 579–610; Math. USSR-Sb. 7, 569–596. MR40#4939.

    Google Scholar 

  • A. Hurwitz, (1891), "Ueber Riemann'sche Flächen mit gegebenen Verzweigungspunkten", Math. Ann. 39, 1–61. FdM23,429.

    Article  MathSciNet  MATH  Google Scholar 

  • A. Hurwitz, (1902), "Ueber die Anzahl der Riemann'schen Flächen mit gegebenen Verzweigungspunkten", Math. Ann. 55, 53–66. FdM32,404.

    Article  MATH  Google Scholar 

  • S. Kinoshita and H. Terasaka, (1957), "On unions of knots", Osaka Math. J. 9, 131–153. MR20#4846.

    MathSciNet  MATH  Google Scholar 

  • H. Levinson, (1973), "Decomposable braids and linkages", Trans. Amer. Math. Soc. 178, 111–126.

    Article  MathSciNet  MATH  Google Scholar 

  • Seymour Lipschutz, (1961), "On a finite matrix representation of the braid group", Arch. Math. 12, 7–12. MR23#A2469.

    Article  MathSciNet  MATH  Google Scholar 

  • Seymour Lipschutz, (1963), "Note on a paper by Shepperd on the braid group", Proc. Amer. Math. Soc. 14, 225–227. MR26#4350.

    Article  MathSciNet  MATH  Google Scholar 

  • Colin Maclachlan, (1973), "On a Conjecture of Magnus on the Hurwitz Monodromy Group", Math. Z. 132, 45–50.

    Article  MathSciNet  MATH  Google Scholar 

  • Wilhelm Magnus, (1934), "Über Automorphismen von Fundamentalgruppen berandeter Flächen", Math. Ann. 109, 617–646. FdM60,91.

    Article  MathSciNet  MATH  Google Scholar 

  • W. Magnus, (1972), "Braids and Riemann surfaces", Comm. Pure Appl. Math. 25, 151–161. Zbl.226.55002.

    Article  MathSciNet  MATH  Google Scholar 

  • Wilhelm Magnus and Ada Peluso, (1967), "On knot groups", Comm. Pure Appl. Math. 20, 749–770. MR36#5930.

    Article  MathSciNet  MATH  Google Scholar 

  • Wilhelm Magnus and Ada Peluso, (1969), "On a theorem of V.I. Arnol'd", Comm. Pure Appl. Math. 22, 683–692. MR41#8658.

    Article  MathSciNet  MATH  Google Scholar 

  • Г.С. Маканин [G.S. Makanin], (1968), "Проблема сопряженности в группе кос" [The conjugacy problem in the braid group], Dokl. Akad. Nauk SSSR 182, 495–496; Soviet Math. Dokl. 9, 1156–1157. MR38#2195.

    MathSciNet  Google Scholar 

  • A. Markoff, (1936), "Über die freie Äquivalenz der geschlossenen Zöpfe", Mat. Sb. (NS) 1 (43), 73–78. FdM62,658.

    MATH  Google Scholar 

  • 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.

    MathSciNet  Google Scholar 

  • K. Murasugi, (1973), "On closed 3-braids", (Preprint).

    Google Scholar 

  • K. Murasugi and R.S.D. Thomas, (1972), "Isotopic closed nonconjugate braids", Proc. Amer. Math. Soc. 33, 137–139. MR45#1149.

    Article  MathSciNet  MATH  Google Scholar 

  • M.H.A. Newman, (1942), "On a string problem of Dirac", J. London Math. Soc. 17, 173–177. MR4,252.

    Article  MathSciNet  MATH  Google Scholar 

  • C.D. Papakyriakopoulos, (1957), "On Dehn's lemma and the asphericity of knots", Ann. of Math. (2) 66, 1–26. MR19,761.

    Article  MathSciNet  MATH  Google Scholar 

  • Kurt Reidemeister, (1960), "Knoten und Geflechte", Nachr. Akad. Wiss. Gottingen Math.-Phys. Kl. 2, 105–115. MR22#1913.

    MathSciNet  MATH  Google Scholar 

  • G.P. Scott, (1970), "Braid groups and the group of homeomorphisms of a surface", Proc. Cambridge Philos. Soc. 68, 605–617. Zbl.203,563.

    Article  MathSciNet  MATH  Google Scholar 

  • J.A.H. Shepperd, (1962), "Braids which can be plaited with their ends tied together at each end", Proc. Roy. soc. London Ser. A 265, 229–244. MR24#A2959.

    Article  MathSciNet  MATH  Google Scholar 

  • Jacques Tits, (1969), "Le problème des mots dans les groupes de Coxeter", Symposia Mathematica, INDAM, Rome, 1967/68, 1, 175–185 (Academic Press, London, New York). MR40#7339.

    Article  MATH  Google Scholar 

  • James Van Buskirk, (1966), "Braid groups of compact 2-manifolds with elements of finite order", Trans. Amer. Math. Soc. 122, 81–97. MR32#6440.

    Article  MathSciNet  MATH  Google Scholar 

  • 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; Math. USSR-Sb. 16 (1972), 223–236. MR45#7701.

    Google Scholar 

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  1. Wilhelm Magnus

    Present address: , 11 Lomond Place, 10804, New Rochelle, New York, USA

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  1. Courant Institute of Mathematical Sciences, New York University, USA

    Wilhelm Magnus

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M. F. Newman

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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/BFb0065203

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