We associate to every positive braid a group, generalizing the geometric monodromy group of an isolated plane curve singularity. If the closure of the braid is a knot, we identify the corresponding group with a framed mapping class group. In particular, this gives a well defined knot invariant. As an application, we obtain that the geometric monodromy group of an irreducible singularity is determined by the genus and the Arf invariant of the associated knot.
Ferretti, Livio 1
@article{10_2140_agt_2025_25_161,
author = {Ferretti, Livio},
title = {On positive braids, monodromy groups and framings},
journal = {Algebraic and Geometric Topology},
pages = {161--205},
year = {2025},
volume = {25},
number = {1},
doi = {10.2140/agt.2025.25.161},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2025.25.161/}
}
Ferretti, Livio. On positive braids, monodromy groups and framings. Algebraic and Geometric Topology, Tome 25 (2025) no. 1, pp. 161-205. doi: 10.2140/agt.2025.25.161
[1] , Sur la monodromie des singularités isolées d’hypersurfaces complexes, Invent. Math. 20 (1973) 147 | DOI
[2] , Le groupe de monodromie du déploiement des singularités isolées de courbes planes, I, Math. Ann. 213 (1975) 1 | DOI
[3] , Generic immersions of curves, knots, monodromy and Gordian number, Inst. Hautes Études Sci. Publ. Math. 88 (1998) 151 | DOI
[4] , Real deformations and complex topology of plane curve singularities, Ann. Fac. Sci. Toulouse Math. 8 (1999) 5 | DOI
[5] , , , Checkerboard graph monodromies, Enseign. Math. 64 (2018) 65 | DOI
[6] , , Secondary braid groups, preprint (2020)
[7] , , Plane algebraic curves, Birkhäuser (1986) | DOI
[8] , , Framed mapping class groups and the monodromy of strata of abelian differentials, J. Eur. Math. Soc. 25 (2023) 4719 | DOI
[9] , , Representative braids for links associated to plane immersed curves, J. Knot Theory Ramifications 9 (2000) 1 | DOI
[10] , Positive braids are visually prime, Proc. Lond. Math. Soc. 67 (1993) 384 | DOI
[11] , , A primer on mapping class groups, 49, Princeton Univ. Press (2012)
[12] , Positive Hopf plumbed links with maximal signature, PhD thesis, Universität Bern (2021)
[13] , , , Lissajous curves as A’Campo divides, torus knots and their fiber surfaces, Tokyo J. Math. 25 (2002) 485 | DOI
[14] , Dynkin diagrams for singularities of functions of two variables, Funkcional. Anal. i Priložen. 8 (1974) 23
[15] , Intersection matrices for certain singularities of functions of two variables, Funkcional. Anal. i Priložen. 8 (1974) 11
[16] , Visualization of A’Campo’s fibered links and unknotting operation, Topology Appl. 121 (2002) 287 | DOI
[17] , Plumbing constructions of connected divides and the Milnor fibers of plane curve singularities, Indag. Math. 13 (2002) 499 | DOI
[18] , Spin structures and quadratic forms on surfaces, J. Lond. Math. Soc. 22 (1980) 365 | DOI
[19] , Generalized braid groups and mapping class groups, J. Knot Theory Ramifications 6 (1997) 715 | DOI
[20] , On the genus defect of positive braid knots, Algebr. Geom. Topol. 20 (2020) 403 | DOI
[21] , Fundamental group of discriminant complements of Brieskorn–Pham polynomials, C. R. Math. Acad. Sci. Paris 345 (2007) 93 | DOI
[22] , Singular points of complex hypersurfaces, 61, Princeton Univ. Press (1968)
[23] , , Groupe de monodromie géométrique des singularités simples, Math. Ann. 306 (1996) 231 | DOI
[24] , , Vanishing cycles, plane curve singularities and framed mapping class groups, Geom. Topol. 25 (2021) 3179 | DOI
[25] , Homology of the moduli spaces and mapping class groups of framed, r–Spin and Pin surfaces, J. Topol. 7 (2014) 155 | DOI
[26] , Curves intersecting in a circuit pattern, Topology Appl. 332 (2023) 108522 | DOI
[27] , Artin groups and geometric monodromy, Invent. Math. 138 (1999) 563 | DOI
[28] , The braid index of an algebraic link, from: "Braids", Contemp. Math. 78, Amer. Math. Soc. (1988) 697 | DOI
Cité par Sources :