Braids in Pau – An Introduction
[Tresses à Pau – une introduction]
Annales Mathématiques Blaise Pascal, Tome 18 (2011) no. 1, pp. 1-14.

Dans ce travail, nous décrivons les liaisons historiques entre l’étude de variétés de dimension 3 (notamment, la théorie de nœuds) et l’étude de la topologie des courbes planes complexes, dont l’accent est posé sur le rôle des groupes de tresses et des invariantes du type Alexander (torsions, différents incarnations des polynômes d’Alexander). Nous finissons en présentant un example avec des calculs détaillés.

In this work, we describe the historic links between the study of 3-dimensional manifolds (specially knot theory) and the study of the topology of complex plane curves with a particular attention to the role of braid groups and Alexander-like invariants (torsions, different instances of Alexander polynomials). We finish with detailed computations in an example.

DOI : https://doi.org/10.5802/ambp.292
Classification : 14H50,  14D05,  57M25,  57C10,  20F36
Mots clés : Nœuds, courbes, torsion, polynômes d’Alexander
@article{AMBP_2011__18_1_1_0,
     author = {Enrique Artal Bartolo and Vincent Florens},
     title = {Braids in {Pau} {\textendash} {An} {Introduction}},
     journal = {Annales Math\'ematiques Blaise Pascal},
     pages = {1--14},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {18},
     number = {1},
     year = {2011},
     doi = {10.5802/ambp.292},
     mrnumber = {2830087},
     zbl = {1214.14001},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.292/}
}
Enrique Artal Bartolo; Vincent Florens. Braids in Pau – An Introduction. Annales Mathématiques Blaise Pascal, Tome 18 (2011) no. 1, pp. 1-14. doi : 10.5802/ambp.292. https://ambp.centre-mersenne.org/articles/10.5802/ambp.292/

[1] J. W. Alexander A lemma on a system of knotted curves, Proc. Nat. Acad. Sci. USA, Volume 9 (1923), pp. 93-95 | Article

[2] J. W. Alexander Topological invariants of knots and links, Trans. Amer. Math. Soc., Volume 30 (1928), pp. 275-306 | Article | MR 1501429

[3] E. Artin Theorie der Zöpfe, Abh. Math. Sem. Univ. Hamburg, Volume 4 (1925), pp. 47-72 | Article

[4] J. Carmona Monodromía de trenzas de curvas algebraicas planas (2003) (Ph. D. Thesis)

[5] Fabrizio Catanese; Bronislaw Wajnryb The 3-cuspidal quartic and braid monodromy of degree 4 coverings, Projective varieties with unexpected properties, Walter de Gruyter GmbH & Co. KG, Berlin, 2005, pp. 113-129 | MR 2202250 | Zbl 1110.14037

[6] A. Cayley A theorem on groups, Math. Ann., Volume 13 (1878) no. 4, pp. 561-565 | Article | MR 1509978

[7] O. Chisini Una suggestiva rappresentazione reale per le curve algebriche piane, Ist. Lombardo, Rend., II. Ser., Volume 66 (1933), pp. 1141-1155 | Zbl 0008.22001

[8] J. I. Cogolludo Braid monodromy of algebraic curve, Ann. Math. Blaise Pascal, Volume 18 (2011) no. 1, pp. 141-209 | Article

[9] J. I. Cogolludo Agustín; V. Florens Twisted Alexander polynomials of plane algebraic curves, J. Lond. Math. Soc. (2), Volume 76 (2007) no. 1, pp. 105-121 | Article | MR 2351611 | Zbl 1151.14022

[10] A. I. Degtyarev A divisibility theorem for the Alexander polynomial of a plane algebraic curve, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), Volume 280 (2001) no. Geom. i Topol. 7, p. 146-156, 300 | Article | MR 1879260 | Zbl 1075.14025

[11] M. Dehn Über die Topologie des dreidimensionalen Raumes, Math. Ann., Volume 69 (1910) no. 1, pp. 137-168 | Article | MR 1511580

[12] M. Dehn Die beiden Kleeblattschlingen, Math. Ann., Volume 75 (1914) no. 3, pp. 402-413 | Article | MR 1511799

[13] Walther Dyck Gruppentheoretische Studien, Math. Ann., Volume 20 (1882) no. 1, pp. 1-44 | Article | MR 1510147

[14] R. Fox; L. Neuwirth The braid groups, Math. Scand., Volume 10 (1962), pp. 119-126 | MR 150755 | Zbl 0117.41101

[15] R. H. Fox On the complementary domains of a certain pair of inequivalent knots, Nederl. Akad. Wetensch. Proc. Ser. A. 55 = Indagationes Math., Volume 14 (1952), pp. 37-40 | MR 48024 | Zbl 0046.16802

[16] Wolfgang Franz Über die Torsion einer Überdeckung., J. Reine Angew. Math., Volume 173 (1935), pp. 245-254 | Article | Zbl 0012.12702

[17] Stefan Friedl; Stefano Vidussi Twisted Alexander polynomials and symplectic structures, Amer. J. Math., Volume 130 (2008) no. 2, pp. 455-484 | Article | MR 2405164 | Zbl 1154.57021

[18] Juan González-Meneses Basic results on braid groups, Ann. Math. Blaise Pascal, Volume 18 (2011) no. 1, pp. 15-59 | Article

[19] A. Hurwitz Ueber Riemann’sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann., Volume 39 (1891) no. 1, pp. 1-60 | Article | MR 1510692

[20] Egbert R. Van Kampen On the Fundamental Group of an Algebraic Curve, Amer. J. Math., Volume 55 (1933) no. 1-4, pp. 255-267 | Article | MR 1506962 | Zbl 0006.41502

[21] Paul Kirk; Charles Livingston Twisted Alexander invariants, Reidemeister torsion, and Casson-Gordon invariants, Topology, Volume 38 (1999) no. 3, pp. 635-661 | Article | MR 1670420 | Zbl 0928.57005

[22] Paul Kirk; Charles Livingston Twisted knot polynomials: inversion, mutation and concordance, Topology, Volume 38 (1999) no. 3, pp. 663-671 | Article | MR 1670424 | Zbl 0928.57006

[23] Teruaki Kitano Twisted Alexander polynomial and Reidemeister torsion, Pacific J. Math., Volume 174 (1996) no. 2, pp. 431-442 http://projecteuclid.org/getRecord?id=euclid.pjm/1102365178 | MR 1405595 | Zbl 0863.57001

[24] Vik. S. Kulikov; M. Taĭkher Braid monodromy factorizations and diffeomorphism types, Izv. Ross. Akad. Nauk Ser. Mat., Volume 64 (2000) no. 2, pp. 89-120 | Article | MR 1770673 | Zbl 1004.14005

[25] A. Libgober Alexander polynomial of plane algebraic curves and cyclic multiple planes, Duke Math. J., Volume 49 (1982) no. 4, pp. 833-851 http://projecteuclid.org/getRecord?id=euclid.dmj/1077315533 | Article | MR 683005 | Zbl 0524.14026

[26] A. Libgober On the homotopy type of the complement to plane algebraic curves, J. Reine Angew. Math., Volume 367 (1986), pp. 103-114 | Article | MR 839126 | Zbl 0576.14019

[27] A. Libgober Invariants of plane algebraic curves via representations of the braid groups, Invent. Math., Volume 95 (1989) no. 1, pp. 25-30 | Article | MR 969412 | Zbl 0674.14015

[28] A. Libgober Characteristic varieties of algebraic curves, Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001) (NATO Sci. Ser. II Math. Phys. Chem.) Volume 36, Kluwer Acad. Publ., Dordrecht, 2001, pp. 215-254 | MR 1866902 | Zbl 1045.14016

[29] Xiao Song Lin Representations of knot groups and twisted Alexander polynomials, Acta Math. Sin. (Engl. Ser.), Volume 17 (2001) no. 3, pp. 361-380 | Article | MR 1852950 | Zbl 0986.57003

[30] Wilhelm Magnus Über Automorphismen von Fundamentalgruppen berandeter Flächen, Math. Ann., Volume 109 (1934) no. 1, pp. 617-646 | Article | MR 1512913 | Zbl 0009.03901

[31] G. Massuyeau An introduction to the abelian Reidemeister torsion of three-dimensional manifolds, Ann. Math. Blaise Pascal, Volume 18 (2011) no. 1, pp. 61-140 | Article

[32] J. Milnor Whitehead torsion, Bull. Amer. Math. Soc., Volume 72 (1966), pp. 358-426 | Article | MR 196736 | Zbl 0147.23104

[33] John Milnor A duality theorem for Reidemeister torsion, Ann. of Math. (2), Volume 76 (1962), pp. 137-147 | Article | MR 141115 | Zbl 0108.36502

[34] B. Moishezon Algebraic surfaces and the arithmetic of braids. I, Arithmetic and geometry, Vol. II (Progr. Math.) Volume 36, Birkhäuser Boston, Boston, MA, 1983, pp. 199-269 | MR 717613 | Zbl 0592.14012

[35] C. D. Papakyriakopoulos On Dehn’s lemma and the asphericity of knots, Ann. of Math. (2), Volume 66 (1957), pp. 1-26 | Article | MR 90053 | Zbl 0078.16402

[36] C. D. Papakyriakopoulos On Dehn’s lemma and the asphericity of knots, Proc. Nat. Acad. Sci. U.S.A., Volume 43 (1957), pp. 169-172 | Article | MR 82671 | Zbl 0078.16401

[37] V. G. Turaev Reidemeister torsion in knot theory, Uspekhi Mat. Nauk, Volume 41 (1986) no. 1(247), p. 97-147, 240 | MR 832411 | Zbl 0602.57005

[38] Vladimir Turaev Introduction to combinatorial torsions, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001 (Notes taken by Felix Schlenk) | MR 1809561 | Zbl 0970.57001

[39] Masaaki Wada Twisted Alexander polynomial for finitely presentable groups, Topology, Volume 33 (1994) no. 2, pp. 241-256 | Article | MR 1273784 | Zbl 0822.57006

[40] Friedhelm Waldhausen On irreducible 3-manifolds which are sufficiently large, Ann. of Math. (2), Volume 87 (1968), pp. 56-88 | Article | MR 224099 | Zbl 0157.30603

[41] W. Wirtinger Über die Verzweigungen bei Funktionen von zwei Veränderlichen, Jahresberichte D. M. V., Volume 14 (1905), 517 pages

[42] W. Wirtinger Zur formalen Theorie der Funktionen von mehr komplexen Veränderlichen, Math. Ann., Volume 97 (1927) no. 1, pp. 357-375 | Article | MR 1512366

[43] Oscar Zariski On the Problem of Existence of Algebraic Functions of Two Variables Possessing a Given Branch Curve, Amer. J. Math., Volume 51 (1929) no. 2, pp. 305-328 | Article | MR 1506719