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Tr. Mosk. Mat. Obs., 2013, Volume 74, Issue 1, Pages 115–173 (Mi mmo542)  

This article is cited in 11 scientific papers (total in 11 papers)

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

I. A. Dynnikovab, M. V. Prasolova

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b Steklov Mathematical Institute of the Russian Academy of Sciences

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.

Key words and phrases: Legendrian knots; monotonic simplification; representation of links by braids.

Funding Agency Grant Number
Russian Foundation for Basic Research 10-01-91056-НЦНИ_а
Ministry of Education and Science of the Russian Federation 2010-220-01-077


Full text: PDF file (1008 kB)
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English version:
Transactions of the Moscow Mathematical Society, 2013, 74, 97–144

Bibliographic databases:

UDC: 515.162.8, 514.763.34
MSC: 57M25, 57R15
Received: 03.04.2012
Revised: 09.03.2013

Citation: I. A. Dynnikov, M. V. Prasolov, “Bypasses for rectangular diagrams. A proof of the Jones conjecture and related questions”, Tr. Mosk. Mat. Obs., 74, no. 1, MCCME, M., 2013, 115–173; Trans. Moscow Math. Soc., 74 (2013), 97–144

Citation in format AMSBIB
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\by I.~A.~Dynnikov, M.~V.~Prasolov
\paper Bypasses for rectangular diagrams. A proof of the Jones conjecture and related questions
\serial Tr. Mosk. Mat. Obs.
\yr 2013
\vol 74
\issue 1
\pages 115--173
\publ MCCME
\publaddr M.
\mathnet{http://mi.mathnet.ru/mmo542}
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\zmath{https://zbmath.org/?q=an:06371557}
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\transl
\jour Trans. Moscow Math. Soc.
\yr 2013
\vol 74
\pages 97--144
\crossref{https://doi.org/10.1090/S0077-1554-2014-00210-7}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84960089959}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Hamer J., Ito T., Kawamuro K., “Positivities of Knots and Links and the Defect of Bennequin Inequality”, Exp. Math.  crossref  mathscinet  isi  scopus
    2. LaFountain D.J., Menasco W.W., “Embedded Annuli and Jones' Conjecture”, Algebr. Geom. Topol., 14:6 (2014), 3589–3601  crossref  mathscinet  zmath  isi  scopus
    3. Ch. Cornwell, L. Ng, S. Sivek, “Obstructions to Lagrangian concordance”, Algebr. Geom. Topol., 16:2 (2016), 797–824  crossref  mathscinet  zmath  isi  scopus
    4. I. A. Dynnikov, M. V. Prasolov, “Rectangular diagrams of surfaces: representability”, Sb. Math., 208:6 (2017), 791–841  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    5. P. Feller, D. Krcatovich, “On cobordisms between knots, braid index, and the upsilon-invariant”, Math. Ann., 369:1-2 (2017), 301–329  crossref  mathscinet  zmath  isi  scopus
    6. Ch. R. Cornwell, “KCH representations, augmentations, and $A$-polynomials”, J. Symplectic Geom., 15:4 (2017), 983–1017  crossref  mathscinet  zmath  isi
    7. K. Hayden, “Minimal braid representatives of quasipositive links”, Pac. J. Math., 295:2 (2018), 421–427  crossref  mathscinet  zmath  isi  scopus
    8. T. Ito, “On a relation between the self-linking number and the braid index of closed braids in open books”, Kyoto J. Math., 58:1 (2018), 193–226  crossref  mathscinet  zmath  isi  scopus
    9. I. A. Dynnikov, “Transverse-Legendrian links”, Sib. elektron. matem. izv., 16 (2019), 1960–1980  mathnet  crossref
    10. Dynnikov I., Prasolov M., “Classification of Legendrian Knots of Topological Type 7(6) With Maximal Thurston-Bennequin Number”, J. Knot Theory Ramifications, 28:14 (2019), 1950089  crossref  mathscinet  zmath  isi  scopus
    11. Feller P., Hubbard D., “Braids With as Many Full Twists as Strands Realize the Braid Index”, J. Topol., 12:4 (2019), 1069–1092  crossref  mathscinet  zmath  isi  scopus
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