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Transactions of the Moscow Mathematical Society
Transactions of the Moscow Mathematical Society
ISSN 1547-738X(online) ISSN 0077-1554(print)

   

 

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


Authors: I. A. Dynnikov and M. V. Prasolov
Translated by:
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 74 (2013), vypusk 1.
Journal: Trans. Moscow Math. Soc. 2013, 97-144
MSC (2010): Primary 57M25; Secondary 57R15
Published electronically: April 9, 2014
MathSciNet review: 3235791
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Abstract | References | Similar Articles | Additional Information

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.


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Additional Information

I. A. Dynnikov
Affiliation: Department of Mechanics and Mathematics, Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow State University
Email: dynnikov@mech.math.msu.su

M. V. Prasolov
Affiliation: Department of Mechanics and Mathematics, Moscow State University
Email: 0x00002a@gmail.com

DOI: http://dx.doi.org/10.1090/S0077-1554-2014-00210-7
Keywords: Legendrian knots, monotonic simplification, representation of links by braids
Published electronically: April 9, 2014
Additional Notes: Supported by the Russian Foundation for Basic Research (Grant no. 10-01-91056-NTsNI_a) and the Russian government (Grant no. 2010-220-01-077).
Article copyright: © Copyright 2014 American Mathematical Society