Knot contact homology, string topology, and the cord algebra

Kai Cieliebak; Tobias Ekholm; Janko Latschev; Lenhard Ng

doi:10.5802/jep.55

Knot contact homology, string topology, and the cord algebra
[Homologie de contact pour les nœuds, topologie des cordes et algèbre des cordes]

Kai Cieliebak ¹ ; Tobias Ekholm ^2,³ ; Janko Latschev ⁴ ; Lenhard Ng ⁵

¹ Institut für Mathematik, Universität Augsburg 86135 Augsburg, Germany
² Department of Mathematics, Uppsala University 751 06 Uppsala, Sweden and
³ Institut Mittag-Leffler Aurav 17, 182 60 Djursholm, Sweden
⁴ Universität Hamburg, Fachbereich Mathematik Bundesstraße 55, 20146 Hamburg, Germany
⁵ Department of Mathematics, Duke University Durham, NC 27708-0320, USA

Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 661-780.

Résumé
Abstract

Le fibré conormal lagrangien $L_{K}$ d’un nœud $K$ dans $ℝ^{3}$ est la sous-variété du fibré cotangent $T^{*} ℝ^{3}$ formée des covecteurs le long de $K$ qui annulent les vecteurs tangents à $K$ . En l’intersectant avec le fibré cotangent unitaire $S^{*} ℝ^{3}$ , on obtient le fibré conormal unitaire $Λ_{K}$ , dont l’homologie de contact legendrienne est un invariant du nœud $K$ , appelé homologie de contact pour les nœuds. Nous définissons une version de la topologie des cordes pour des cordes dans $ℝ^{3} \cup L_{K}$ et montrons qu’elle est isomorphe en degré $0$ à l’homologie de contact pour les nœuds. La topologie des cordes permet une approche topologique de l’algèbre des cordes (qui est aussi isomorphe à l’homologie de contact pour les nœuds en degré $0$ ) et la relie au groupe du nœud. Ceci donne, joint à cet isomorphisme, une nouvelle démonstration du fait que l’homologie de contact pour les nœuds détecte le nœud trivial. Nos techniques font intervenir une analyse détaillée de certains espaces de modules de disques holomorphes dans $T^{*} ℝ^{3}$ avec bord dans $ℝ^{3} \cup L_{K}$ .

The conormal Lagrangian $L_{K}$ of a knot $K$ in $ℝ^{3}$ is the submanifold of the cotangent bundle $T^{*} ℝ^{3}$ consisting of covectors along $K$ that annihilate tangent vectors to $K$ . By intersecting with the unit cotangent bundle $S^{*} ℝ^{3}$ , one obtains the unit conormal $Λ_{K}$ , and the Legendrian contact homology of $Λ_{K}$ is a knot invariant of $K$ , known as knot contact homology. We define a version of string topology for strings in $ℝ^{3} \cup L_{K}$ and prove that this is isomorphic in degree $0$ to knot contact homology. The string topology perspective gives a topological derivation of the cord algebra (also isomorphic to degree $0$ knot contact homology) and relates it to the knot group. Together with the isomorphism this gives a new proof that knot contact homology detects the unknot. Our techniques involve a detailed analysis of certain moduli spaces of holomorphic disks in $T^{*} ℝ^{3}$ with boundary on $ℝ^{3} \cup L_{K}$ .

Reçu le : 2016-02-04
Accepté le : 2017-05-22
Publié le : 2017-06-08

MR Zbl

DOI : 10.5802/jep.55

Classification : 53D42, 55P50, 57R17, 57M27
Keywords: Holomorphic curve, string topology, conormal bundle, knot invariant, Lagrangian submanifold, Legendrian submanifold
Mot clés : Courbe holomorphe, topologie des cordes, fibré conormal, invariant de nœud, sous-variété lagrangienne, sous-variété legendrienne

Affiliations des auteurs :

Kai Cieliebak ¹ ; Tobias Ekholm ^{2, 3} ; Janko Latschev ⁴ ; Lenhard Ng ⁵

¹ Institut für Mathematik, Universität Augsburg 86135 Augsburg, Germany
² Department of Mathematics, Uppsala University 751 06 Uppsala, Sweden and
³ Institut Mittag-Leffler Aurav 17, 182 60 Djursholm, Sweden
⁴ Universität Hamburg, Fachbereich Mathematik Bundesstraße 55, 20146 Hamburg, Germany
⁵ Department of Mathematics, Duke University Durham, NC 27708-0320, USA

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{JEP_2017__4__661_0,
     author = {Kai Cieliebak and Tobias Ekholm and Janko Latschev and Lenhard Ng},
     title = {Knot contact homology, string topology, and~the cord algebra},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {661--780},
     publisher = {\'Ecole polytechnique},
     volume = {4},
     year = {2017},
     doi = {10.5802/jep.55},
     zbl = {1380.53101},
     mrnumber = {3665612},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.55/}
}

TY  - JOUR
AU  - Kai Cieliebak
AU  - Tobias Ekholm
AU  - Janko Latschev
AU  - Lenhard Ng
TI  - Knot contact homology, string topology, and the cord algebra
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2017
SP  - 661
EP  - 780
VL  - 4
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.55/
DO  - 10.5802/jep.55
LA  - en
ID  - JEP_2017__4__661_0
ER  -

%0 Journal Article
%A Kai Cieliebak
%A Tobias Ekholm
%A Janko Latschev
%A Lenhard Ng
%T Knot contact homology, string topology, and the cord algebra
%J Journal de l’École polytechnique — Mathématiques
%D 2017
%P 661-780
%V 4
%I École polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.55/
%R 10.5802/jep.55
%G en
%F JEP_2017__4__661_0

Kai Cieliebak; Tobias Ekholm; Janko Latschev; Lenhard Ng. Knot contact homology, string topology, and the cord algebra. Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 661-780. doi : 10.5802/jep.55. https://jep.centre-mersenne.org/articles/10.5802/jep.55/

Bibliographie
Cité par

[1] M. Aganagic, T. Ekholm, L. Ng & C. Vafa - “Topological strings, D-model, and knot contact homology”, Adv. Theo. Math. Phys. 18 (2014) no. 4, p. 827-956 | DOI | MR | Zbl

[2] S. Basu, J. McGibbon, D. Sullivan & M. Sullivan - “Transverse string topology and the cord algebra”, J. Symplectic Geom. 13 (2015) no. 1, p. 1-16 | DOI | MR | Zbl

[3] F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki & E. Zehnder - “Compactness results in symplectic field theory”, Geom. Topol. 7 (2003), p. 799-888 | DOI | MR | Zbl

[4] M. Chas & D. Sullivan - “String topology” (1999), arXiv:math.GT/9911159

[5] K. Cieliebak, T. Ekholm & J. Latschev - “Compactness for holomorphic curves with switching Lagrangian boundary conditions”, J. Symplectic Geom. 8 (2010) no. 3, p. 267-298 | DOI | MR | Zbl

[6] K. Cieliebak & J. Latschev - “The role of string topology in symplectic field theory”, in New perspectives and challenges in symplectic field theory, CRM Proc. Lecture Notes, vol. 49, American Mathematical Society, Providence, RI, 2009, p. 113-146 | DOI | MR | Zbl

[7] G. Dimitroglou Rizell - “Lifting pseudo-holomorphic polygons to the symplectisation of $P \times ℝ$ and applications”, Quantum Topol. 7 (2016) no. 1, p. 29-105 | DOI | MR | Zbl

[8] N. M. Dunfield & S. Garoufalidis - “Non-triviality of the $A$ -polynomial for knots in $S^{3}$ ”, Algebraic Geom. Topol. 4 (2004), p. 1145-1153 | DOI | MR | Zbl

[9] T. Ekholm - “Morse flow trees and Legendrian contact homology in 1-jet spaces”, Geom. Topol. 11 (2007), p. 1083-1224 | DOI | MR | Zbl

[10] T. Ekholm - “Rational symplectic field theory over $ℤ_{2}$ for exact Lagrangian cobordisms”, J. Eur. Math. Soc. (JEMS) 10 (2008) no. 3, p. 641-704 | DOI | Zbl

[11] T. Ekholm, J. Etnyre, L. Ng & M. Sullivan - “Filtrations on the knot contact homology of transverse knots”, Math. Ann. 355 (2013) no. 4, p. 1561-1591 | DOI | MR | Zbl

[12] T. Ekholm, J. Etnyre, L. Ng & M. Sullivan - “Knot contact homology”, Geom. Topol. 17 (2013) no. 2, p. 975-1112 | DOI | MR | Zbl

[13] T. Ekholm, J. Etnyre & M. Sullivan - “The contact homology of Legendrian submanifolds in $ℝ^{2 n + 1}$ ”, J. Differential Geom. 71 (2005) no. 2, p. 177-305 | DOI | MR | Zbl

[14] T. Ekholm, J. Etnyre & M. Sullivan - “Orientations in Legendrian contact homology and exact Lagrangian immersions”, Internat. J. Math. 16 (2005) no. 5, p. 453-532 | DOI | MR | Zbl

[15] T. Ekholm, J. Etnyre & M. Sullivan - “Legendrian contact homology in $P \times ℝ$ ”, Trans. Amer. Math. Soc. 359 (2007) no. 7, p. 3301-3335 | DOI | MR | Zbl

[16] T. Ekholm & T. Kálmán - “Isotopies of Legendrian 1-knots and Legendrian 2-tori”, J. Symplectic Geom. 6 (2008) no. 4, p. 407-460 | DOI | MR | Zbl

[17] T. Ekholm, L. Ng & V. Shende - “A complete knot invariant from contact homology”, arXiv:1606.07050 | DOI | Zbl

[18] T. Ekholm & I. Smith - “Exact Lagrangian immersions with one double point revisited”, Math. Ann. 358 (2014) no. 1-2, p. 195-240 | DOI | MR | Zbl

[19] T. Ekholm & I. Smith - “Exact Lagrangian immersions with a single double point”, J. Amer. Math. Soc. 29 (2016) no. 1, p. 1-59 | DOI | MR | Zbl

[20] Y. Eliashberg, A. Givental & H. Hofer - “Introduction to symplectic field theory”, Geom. Funct. Anal. (2000), p. 560-673, Special Volume, Part II, GAFA 2000 (Tel Aviv, 1999) | MR | Zbl

[21] A. Floer - “Morse theory for Lagrangian intersections”, J. Differential Geom. 28 (1988) no. 3, p. 513-547 | DOI | MR | Zbl

[22] K. Fukaya, Y.-G. Oh, H. Ohta & K. Ono - Lagrangian intersection Floer theory: anomaly and obstruction. Part I, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI, 2009 | MR | Zbl

[23] C. Gordon & T. Lidman - “Knot contact homology detects cabled, composite, and torus knots”, arXiv:1509.01642 | DOI | Zbl

[24] C. Gordon & J. Luecke - “Knots are determined by their complements”, J. Amer. Math. Soc. 2 (1989) no. 2, p. 371-415 | DOI | MR | Zbl

[25] M. W. Hirsch - Differential topology, Graduate Texts in Math., vol. 33, Springer-Verlag, New York-Heidelberg, 1976 | MR | Zbl

[26] - The Floer memorial volume (H. Hofer, C. H. Taubes, A. Weinstein & E. Zehnder, eds.), Progress in Mathematics, vol. 133, Birkhäuser Verlag, Basel, 1995 | MR | Zbl

[27] H. Hofer, K. Wysocki & E. Zehnder - “A general Fredholm theory. I. A splicing-based differential geometry”, J. Eur. Math. Soc. (JEMS) 9 (2007) no. 4, p. 841-876 | DOI | MR | Zbl

[28] P. B. Kronheimer & T. S. Mrowka - “Dehn surgery, the fundamental group and SU $(2)$ ”, Math. Res. Lett. 11 (2004) no. 5-6, p. 741-754 | DOI | MR | Zbl

[29] R. H. Lagrange & A. H. Rhemtulla - “A remark on the group rings of order preserving permutation groups.”, Canad. Math. Bull. 11 (1968), p. 679-680 | DOI | MR | Zbl

[30] L. Ng - “Knot and braid invariants from contact homology. I”, Geom. Topol. 9 (2005), p. 247-297 | DOI | MR | Zbl

[31] L. Ng - “Knot and braid invariants from contact homology. II”, Geom. Topol. 9 (2005), p. 1603-1637, With an appendix by the author and Siddhartha Gadgil | DOI | MR | Zbl

[32] L. Ng - “Framed knot contact homology”, Duke Math. J. 141 (2008) no. 2, p. 365-406 | DOI | MR | Zbl

[33] L. Ng - “Combinatorial knot contact homology and transverse knots”, Adv. Math. 227 (2011) no. 6, p. 2189-2219 | DOI | MR | Zbl

[34] L. Ng - “A topological introduction to knot contact homology”, in Contact and symplectic topology, Bolyai Soc. Math. Stud., vol. 26, János Bolyai Math. Soc., Budapest, 2014, p. 485-530 | DOI | MR | Zbl

[35] V. Shende - “The conormal torus is a complete knot invariant”, arXiv:1604.03520

[36] D. Sullivan - “Open and closed string field theory interpreted in classical algebraic topology”, in Topology, geometry and quantum field theory, London Math. Soc. Lecture Note Ser., vol. 308, Cambridge Univ. Press, Cambridge, 2004, p. 344-357 | DOI | MR | Zbl

[37] D. Sullivan - “String topology background and present state”, in Current developments in mathematics, 2005, Int. Press, Somerville, MA, 2007, p. 41-88 | Zbl

Cité par Sources :