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

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-ÍÖÍÈ_à Ministry of Education and Science of the Russian Federation 2010-220-01-077

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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
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
\Bibitem{DynPra13} \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} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3235791} \zmath{https://zbmath.org/?q=an:06371557} \elib{https://elibrary.ru/item.asp?id=21369365} \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.
2. LaFountain D.J., Menasco W.W., “Embedded Annuli and Jones' Conjecture”, Algebr. Geom. Topol., 14:6 (2014), 3589–3601
3. Ch. Cornwell, L. Ng, S. Sivek, “Obstructions to Lagrangian concordance”, Algebr. Geom. Topol., 16:2 (2016), 797–824
4. I. A. Dynnikov, M. V. Prasolov, “Rectangular diagrams of surfaces: representability”, Sb. Math., 208:6 (2017), 791–841
5. P. Feller, D. Krcatovich, “On cobordisms between knots, braid index, and the upsilon-invariant”, Math. Ann., 369:1-2 (2017), 301–329
6. Ch. R. Cornwell, “KCH representations, augmentations, and $A$-polynomials”, J. Symplectic Geom., 15:4 (2017), 983–1017
7. K. Hayden, “Minimal braid representatives of quasipositive links”, Pac. J. Math., 295:2 (2018), 421–427
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
9. I. A. Dynnikov, “Transverse-Legendrian links”, Sib. elektron. matem. izv., 16 (2019), 1960–1980
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
11. Feller P., Hubbard D., “Braids With as Many Full Twists as Strands Realize the Braid Index”, J. Topol., 12:4 (2019), 1069–1092
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