Annales Mathématiques
Blaise Pascal
no. 1
Braids in Pau – An Introduction
[Tresses à Pau – une introduction]
Enrique Artal Bartolo1 ; Vincent Florens2
1 Departamento de Matemáticas, IUMA, Facultad de Ciencias, Universidad de Zaragoza, c/ Pedro Cerbuna, 12, 50009 Zaragoza, Spain
2 Laboratoire de Mathématiques et de leurs Applications - PAU UMR CNRS 5142 Bâtiment IPRA - Université de Pau et des Pays de l’Adour Avenue de l’Université - BP 1155 64013 PAU CEDEX, France
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 : 10.5802/ambp.292
Classification : 14H50, 14D05, 57M25, 57C10, 20F36
Keywords: Knots, curves, braid groups, torsion, Alexander polynomial
Mot clés : Nœuds, courbes, torsion, polynômes d’Alexander

Enrique Artal Bartolo 1 ; Vincent Florens 2

1 Departamento de Matemáticas, IUMA, Facultad de Ciencias, Universidad de Zaragoza, c/ Pedro Cerbuna, 12, 50009 Zaragoza, Spain
2 Laboratoire de Mathématiques et de leurs Applications - PAU UMR CNRS 5142 Bâtiment IPRA - Université de Pau et des Pays de l’Adour Avenue de l’Université - BP 1155 64013 PAU CEDEX, France 
@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/}
}
TY  - JOUR
AU  - Enrique Artal Bartolo
AU  - Vincent Florens
TI  - Braids in Pau – An Introduction
JO  - Annales mathématiques Blaise Pascal
PY  - 2011
SP  - 1
EP  - 14
VL  - 18
IS  - 1
PB  - Annales mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.292/
DO  - 10.5802/ambp.292
LA  - en
ID  - AMBP_2011__18_1_1_0
ER  - 
%0 Journal Article
%A Enrique Artal Bartolo
%A Vincent Florens
%T Braids in Pau – An Introduction
%J Annales mathématiques Blaise Pascal
%D 2011
%P 1-14
%V 18
%N 1
%I Annales mathématiques Blaise Pascal
%U https://ambp.centre-mersenne.org/articles/10.5802/ambp.292/
%R 10.5802/ambp.292
%G en
%F AMBP_2011__18_1_1_0
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 | DOI

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

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

[4] J. Carmona Monodromía de trenzas de curvas algebraicas planas, Universidad de Zaragoza (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 | Zbl

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

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

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

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

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

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

[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 | Zbl

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

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

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

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

[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 | DOI | MR | Zbl

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

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

[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 | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

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

[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 | Zbl

[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 | DOI | MR | Zbl

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

[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 | DOI

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

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

[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 | Zbl

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

[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 | DOI | MR | Zbl

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

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

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

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

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

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

[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 | DOI | MR

Cité par Sources :