Quasipositive links and Stein surfaces

Hayden, Kyle¹

¹ Columbia University, New York, NY, United States

Geometry & topology, Tome 25 (2021) no. 3, pp. 1441-1477.

Résumé

We study the generalization of quasipositive links from the $3$–sphere to arbitrary closed, orientable $3$–manifolds. Our main result shows that the boundary of any smooth, properly embedded complex curve in a Stein domain is a quasipositive link. This generalizes a result due to Boileau and Orevkov, and it provides the first half of a topological characterization of links in $3$–manifolds which bound complex curves in a Stein filling. Our arguments replace pseudoholomorphic curve techniques with a study of characteristic and open book foliations on surfaces in $3$– and $4$–manifolds.

Détail

DOI : 10.2140/gt.2021.25.1441

Classification : 57R17, 32Q28, 57M25
Keywords: Stein surfaces, complex curves, contact structures, open books, braids, quasipositive links, transverse links

Bibliographie

[1] J A Baldwin, D S Vela-Vick, V Vértesi, On the equivalence of Legendrian and transverse invariants in knot Floer homology, Geom. Topol. 17 (2013) 925 | DOI

[2] I Baykur, J Etnyre, M Hedden, K Kawamuro, J V Horn-Morris, Contact and symplectic geometry and the mapping class groups, report of 2nd SQuaRE meeting (2015)

[3] D Bennequin, Entrelacements et équations de Pfaff, from: "Third Schnepfenried geometry conference, I" (editors D Bernard, T Hangan, R Lutz), Astérisque 107, Soc. Math. France (1983) 87

[4] J S Birman, E Finkelstein, Studying surfaces via closed braids, J. Knot Theory Ramifications 7 (1998) 267 | DOI

[5] J S Birman, W W Menasco, Stabilization in the braid groups, II : Transversal simplicity of knots, Geom. Topol. 10 (2006) 1425 | DOI

[6] M Boileau, S Orevkov, Quasi-positivité d’une courbe analytique dans une boule pseudo-convexe, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 825 | DOI

[7] K Cieliebak, Y Eliashberg, From Stein to Weinstein and back, 59, Amer. Math. Soc. (2012) | DOI

[8] Y Eliashberg, Topological characterization of Stein manifolds of dimension > 2, Int. J. Math. 1 (1990) 29 | DOI

[9] J B Etnyre, Lectures on open book decompositions and contact structures, from: "Floer homology, gauge theory, and low-dimensional topology" (editors D A Ellwood, P S Ozsváth, A I Stipsicz, Z Szabó), Clay Math. Proc. 5, Amer. Math. Soc. (2006) 103

[10] J B Etnyre, J Van Horn-Morris, Monoids in the mapping class group, from: "Interactions between low-dimensional topology and mapping class groups" (editors R I Baykur, J Etnyre, U Hamenstädt), Geom. Topol. Monogr. 19, Geom. Topol. (2015) 319 | DOI

[11] B Farb, D Margalit, A primer on mapping class groups, 49, Princeton Univ. Press (2012)

[12] T Fiedler, Complex plane curves in the ball, Invent. Math. 95 (1989) 479 | DOI

[13] D Gabai, The Murasugi sum is a natural geometric operation, from: "Low-dimensional topology" (editor S J Lomonaco Jr.), Contemp. Math. 20, Amer. Math. Soc. (1983) 131 | DOI

[14] S Gadgil, D Kulkarni, Relative symplectic caps, 4–genus and fibered knots, Proc. Indian Acad. Sci. Math. Sci. 126 (2016) 261 | DOI

[15] H Geiges, An introduction to contact topology, 109, Cambridge Univ. Press (2008) | DOI

[16] E Giroux, Géométrie de contact : de la dimension trois vers les dimensions supérieures, from: "Proceedings of the International Congress of Mathematicians, II" (editor T Li), Higher Ed. (2002) 405

[17] R E Gompf, Handlebody construction of Stein surfaces, Ann. of Math. 148 (1998) 619 | DOI

[18] M Hedden, Knot theory and algebraic curves in subcritical Stein domains, in preparation

[19] K Honda, W H Kazez, G Matić, The contact invariant in sutured Floer homology, Invent. Math. 176 (2009) 637 | DOI

[20] T Ito, K Kawamuro, Open book foliation, Geom. Topol. 18 (2014) 1581 | DOI

[21] T Ito, K Kawamuro, The defect of the Bennequin–Eliashberg inequality and Bennequin surfaces, Indiana Univ. Math. J. 68 (2019) 799 | DOI

[22] D Mcduff, Singularities and positivity of intersections of J–holomorphic curves, from: "Holomorphic curves in symplectic geometry" (editors M Audin, J Lafontaine), Progr. Math. 117, Birkhäuser (1994) 191 | DOI

[23] M J Micallef, B White, The structure of branch points in minimal surfaces and in pseudoholomorphic curves, Ann. of Math. 141 (1995) 35 | DOI

[24] J Milnor, Morse theory, 51, Princeton Univ. Press (1963)

[25] S Y Orevkov, V V Shevchishin, Markov theorem for transversal links, J. Knot Theory Ramifications 12 (2003) 905 | DOI

[26] E Pavelescu, Braiding knots in contact 3–manifolds, Pacific J. Math. 253 (2011) 475 | DOI

[27] L Rudolph, Algebraic functions and closed braids, Topology 22 (1983) 191 | DOI

[28] L Rudolph, A congruence between link polynomials, Math. Proc. Cambridge Philos. Soc. 107 (1990) 319 | DOI

[29] L Rudolph, Quasipositivity as an obstruction to sliceness, Bull. Amer. Math. Soc. 29 (1993) 51 | DOI

[30] S Smale, On gradient dynamical systems, Ann. of Math. 74 (1961) 199 | DOI

[31] W P Thurston, H E Winkelnkemper, On the existence of contact forms, Proc. Amer. Math. Soc. 52 (1975) 345 | DOI

[32] A Weinstein, Contact surgery and symplectic handlebodies, Hokkaido Math. J. 20 (1991) 241 | DOI

[33] N C Wrinkle, The Markov theorem for transverse knots, PhD thesis, Columbia University (2002)

[34] R Yamamoto, Open books supporting overtwisted contact structures and the Stallings twist, J. Math. Soc. Japan 59 (2007) 751 | DOI