Bypasses for rectangular diagrams. A proof of the Jones conjecture and related questions

I. A. Dynnikov ; M. V. Prasolov

Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 74 (2013) no. 1, pp. 115-173.

Résumé

We give a criterion, in terms of Legendrian knots, for a rectangular diagram to admit a simplification and show that simplifications of two different types are, in a sense, independent of each other. We show that a minimal rectangular diagram maximizes the Thurston–Bennequin number for the corresponding Legendrian links. We prove the Jones conjecture on the invariance of the algebraic number of crossings of a minimal braid representing a given link. We also give a new proof of the monotonic simplification theorem for the unknot.

Bibliographie

[1] Alexander J., “A lemma on a system of knotted curves”, Proc. Nat. Acad. Sci. USA, 9 (1923), 93–95 | DOI | Zbl

[2] Bennequin D., “Entrelacements et équations de Pfaff”, Astérisque, 107–108 (1983), 87–161 | MR | Zbl

[3] Birman J., Braids, links and mapping class groups, Ann. of Math. Studies, 82, Prinston University Press, Prinston, 1975 | MR | Zbl

[4] Birman J., Menasco W., “Studying links via closed braids. IV: Composite links and split links”, Invent. Math., 102 (1990), 115–139 | DOI | MR | Zbl

[5] Birman J., Menasco W., “Studying links via closed braids. V: Closed braid representatives of the unlink”, Trans. AMS, 329:2 (1992), 585–606 | MR | Zbl

[6] Birman J. S., Wrinkle N. C., “On transversally simple knots”, J. Differential Geom., 55:2 (2000), 325–354 | MR | Zbl

[7] Cromwell P., “Embedding knots and links in an open book. I: Basic properties”, Topol. Appl., 64 (1995), 37–58 | DOI | MR | Zbl

[8] Dynnikov I., “Arc-presentations of links: Monotonic simplification”, Fund. Math., 190 (2006), 29–76 | DOI | MR | Zbl

[9] Eliashberg Y., Fraser M., “Classification of topologically trivial Legendrian knots”, CRM Proc. Lecture Notes, 15, no. 15, 1998, 17–51 | MR | Zbl

[10] Eliashberg Y., Fraser M., “Topologically trivial Legendrian knots”, J. Symplectic Geom., 7:2 (2009), 77–127 | DOI | MR | Zbl

[11] Erlandsson T., Geometry of contact transformations in dimension 3, Doctoral Dissertation, Uppsala, 1981

[12] Etnyre J., “Transversal torus knots”, Geom. Topol., 3 (1999), 253–268 | DOI | MR | Zbl

[13] Honda K., “On the classification of tight contact structures, I”, Geom. Topol., 4 (2000), 309–368 | DOI | MR | Zbl

[14] Jones V. F. R., “Hecke algebra representations of braid group and link polynomials”, Ann. of Math., 126:2 (1987), 335–388 | DOI | MR | Zbl

[15] Kawamuro K., “The algebraic crossing number and the braid index of knots and links”, Algeb. Geom. Topol., 6 (2006), 2313–2350 | DOI | MR | Zbl

[16] Malešič J., Traczyk P., “Seifert circles, braid index and the algebraic crossing number”, Topol. Appl., 153:2–3 (2005), 303–317 | MR | Zbl

[17] Markoff A., “Über die freie Äquivalenz der geschlossenen Zöpfe”, Matem. sb., 1 (43):1 (1936), 73–78

[18] Matsuda H., Menasco W., On rectangular diagrams, Legendrian knots and transverse knots, arXiv: 0708.2406v1

[19] Ng L., On arc index and maximal Thurston–Bennequin number, arXiv: math/0612356 | MR

[20] Ng L., Thurston D., “Grid diagrams, braids, and contact geometry”, Proceedings of Gökova Geometry–Topology Conference (2008), 120–136 | MR | Zbl

[21] Orevkov S. Yu., Shevchishin V. V., “Markov theorem for transversal links”, J. Knot Theory Ramifications, 12:7 (2003), 905–913, arXiv: math.GT/0112207 | DOI | MR | Zbl

[22] Ozsváth P., Szabó Z., Thurston D., “Legendrian knots, transverse knots and combinatorial Floer homology”, Geom. Topol., 12 (2008), 941–980 | DOI | MR | Zbl

[23] Świa̧tkowski J., “On the isotopy of Legendrian knots”, Ann. Global Anal. Geom., 10 (1992), 195–207 | DOI | MR | Zbl

[24] Wrinkle N. C., The Markov theorem for transverse knots, Thesis Ph.D., Columbia University, 2002, 51 pp., arXiv: math.GT/0202055 | MR