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

Dynnikov, I.; Prasolov, M.

doi:10.1090/S0077-1554-2014-00210-7

Bypasses for rectangular diagrams. A proof of the Jones conjecture and related questions
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by I. A. Dynnikov and M. V. Prasolov

Trans. Moscow Math. Soc. 2013, 97-144

DOI: https://doi.org/10.1090/S0077-1554-2014-00210-7

Published electronically: April 9, 2014

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Abstract:

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.

References

J. Alexander, A lemma on a system of knotted curves, Proc. Nat. Acad. Sci. USA 9 (1923), 93-95.
Daniel Bennequin, Entrelacements et équations de Pfaff, Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982) Astérisque, vol. 107, Soc. Math. France, Paris, 1983, pp. 87–161 (French). MR 753131
Joan S. Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies, No. 82, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. MR 0375281
Joan S. Birman and William W. Menasco, Studying links via closed braids. IV. Composite links and split links, Invent. Math. 102 (1990), no. 1, 115–139. MR 1069243, DOI 10.1007/BF01233423
Joan S. Birman and William W. Menasco, Studying links via closed braids. V. The unlink, Trans. Amer. Math. Soc. 329 (1992), no. 2, 585–606. MR 1030509, DOI 10.1090/S0002-9947-1992-1030509-1
Joan S. Birman and Nancy C. Wrinkle, On transversally simple knots, J. Differential Geom. 55 (2000), no. 2, 325–354. MR 1847313
Peter R. Cromwell, Embedding knots and links in an open book. I. Basic properties, Topology Appl. 64 (1995), no. 1, 37–58. MR 1339757, DOI 10.1016/0166-8641(94)00087-J
I. A. Dynnikov, Arc-presentations of links: monotonic simplification, Fund. Math. 190 (2006), 29–76. MR 2232855, DOI 10.4064/fm190-0-3
Yakov Eliashberg and Maia Fraser, Classification of topologically trivial Legendrian knots, Geometry, topology, and dynamics (Montreal, PQ, 1995) CRM Proc. Lecture Notes, vol. 15, Amer. Math. Soc., Providence, RI, 1998, pp. 17–51. MR 1619122, DOI 10.1090/crmp/015/02
Yakov Eliashberg and Maia Fraser, Topologically trivial Legendrian knots, J. Symplectic Geom. 7 (2009), no. 2, 77–127. MR 2496415
T. Erlandsson, Geometry of contact transformations in dimension 3, Doctoral Dissertation, Uppsala, 1981.
John B. Etnyre, Transversal torus knots, Geom. Topol. 3 (1999), 253–268. MR 1714912, DOI 10.2140/gt.1999.3.253
Ko Honda, On the classification of tight contact structures. I, Geom. Topol. 4 (2000), 309–368. MR 1786111, DOI 10.2140/gt.2000.4.309
V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, 335–388. MR 908150, DOI 10.2307/1971403
Keiko Kawamuro, The algebraic crossing number and the braid index of knots and links, Algebr. Geom. Topol. 6 (2006), 2313–2350. MR 2286028, DOI 10.2140/agt.2006.6.2313
Jože Malešič and PawełTraczyk, Seifert circles, braid index and the algebraic crossing number, Topology Appl. 153 (2005), no. 2-3, 303–317. MR 2175354, DOI 10.1016/j.topol.2003.05.010
A. Markoff, Über die freie Äquivalenz der geschlossenen Zöpfe, Mat. Sb. 1 (1936), no. 43. no. 1. 73-78.
H. Matsuda and W. Menasco, On rectangular diagrams, Legendrian knots and transverse knots, Preprint, arxiv:0708.2406v1.
Lenhard Ng, On arc index and maximal Thurston-Bennequin number, J. Knot Theory Ramifications 21 (2012), no. 4, 1250031, 11. MR 2890458, DOI 10.1142/S0218216511009820
Lenhard Ng and Dylan Thurston, Grid diagrams, braids, and contact geometry, Proceedings of Gökova Geometry-Topology Conference 2008, Gökova Geometry/Topology Conference (GGT), Gökova, 2009, pp. 120–136. MR 2500576
S. Yu. Orevkov and V. V. Shevchishin, Markov theorem for transversal links, J. Knot Theory Ramifications 12 (2003), no. 7, 905–913. MR 2017961, DOI 10.1142/S0218216503002846
Peter Ozsváth, Zoltán Szabó, and Dylan Thurston, Legendrian knots, transverse knots and combinatorial Floer homology, Geom. Topol. 12 (2008), no. 2, 941–980. MR 2403802, DOI 10.2140/gt.2008.12.941
Jacek Świątkowski, On the isotopy of Legendrian knots, Ann. Global Anal. Geom. 10 (1992), no. 3, 195–207. MR 1186009, DOI 10.1007/BF00136863
Nancy Court Wrinkle, The Markov theorem for transverse knots, ProQuest LLC, Ann Arbor, MI, 2002. Thesis (Ph.D.)–Columbia University. MR 2703285