Knot and braid invariants from contact homology I

Ng, Lenhard¹

¹ Department of Mathematics, Stanford University, Stanford, California 94305, USA

Geometry & topology, Tome 9 (2005) no. 1, pp. 247-297.

Résumé

We introduce topological invariants of knots and braid conjugacy classes, in the form of differential graded algebras, and present an explicit combinatorial formulation for these invariants. The algebras conjecturally give the relative contact homology of certain Legendrian tori in five-dimensional contact manifolds. We present several computations and derive a relation between the knot invariant and the determinant.

Détail

DOI : 10.2140/gt.2005.9.247

Keywords: contact homology, knot invariant, braid representation, differential graded algebra

Bibliographie

[1] J S Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies 82, Princeton University Press (1974)

[2] G Burde, H Zieschang, Knots, de Gruyter Studies in Mathematics 5, Walter de Gruyter Co. (2003)

[3] Y Chekanov, Differential algebra of Legendrian links, Invent. Math. 150 (2002) 441

[4] J H Conway, An enumeration of knots and links, and some of their algebraic properties, from: "Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967)", Pergamon (1970) 329

[5] T Ekholm, J Etnyre, M Sullivan, Legendrian submanifolds in $R^{2n+1}$ and contact homology

[6] Y Eliashberg, Invariants in contact topology, from: "Proceedings of the International Congress of Mathematicians, Vol II (Berlin, 1998)" (1998) 327

[7] Y Eliashberg, A Givental, H Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. (2000) 560

[8] J Epstein, D Fuchs, M Meyer, Chekanov–Eliashberg invariants and transverse approximations of Legendrian knots, Pacific J. Math. 201 (2001) 89

[9] J B Etnyre, L L Ng, J M Sabloff, Invariants of Legendrian knots and coherent orientations, J. Symplectic Geom. 1 (2002) 321

[10] K Fukaya, Y G Oh, Zero-loop open strings in the cotangent bundle and Morse homotopy, Asian J. Math. 1 (1997) 96

[11] J Helton, R Miller, M Stankus, NCAlgebra: a Mathematica package for doing non commuting algebra

[12] S P Humphries, An approach to automorphisms of free groups and braids via transvections, Math. Z. 209 (1992) 131

[13] M Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359

[14] D Krammer, The braid group $B_4$ is linear, Invent. Math. 142 (2000) 451

[15] E S Lee, An endomorphism of the Khovanov invariant, Adv. Math. 197 (2005) 554

[16] W Magnus, Rings of Fricke characters and automorphism groups of free groups, Math. Z. 170 (1980) 91

[17] P Melvin, S Shrestha, The nonuniqueness of Chekanov polynomials of Legendrian knots, Geom. Topol. 9 (2005) 1221

[18] L L Ng, Computable Legendrian invariants, Topology 42 (2003) 55

[19] L Ng, Knot and braid invariants from contact homology II, Geom. Topol. 9 (2005) 1603

[20] H Ooguri, C Vafa, Knot invariants and topological strings, Nuclear Phys. B 577 (2000) 419

[21] P Ozsváth, Z Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004) 58

[22] D Rolfsen, Knots and links, Mathematics Lecture Series 7, Publish or Perish (1976)

[23] J M Sabloff, Invariants of Legendrian knots in circle bundles, Commun. Contemp. Math. 5 (2003) 569