Geodesic
Bypasses for rectangular diagrams. A proof of the Jones conjecture and related questions
Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 74 (2013) no. 1, pp. 115-173

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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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     title = {Bypasses for rectangular diagrams. {A} proof of the {Jones} conjecture and related questions},
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I. A. Dynnikov; M. V. Prasolov. Bypasses for rectangular diagrams. A proof of the Jones conjecture and related questions. Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 74 (2013) no. 1, pp. 115-173. http://geodesic.mathdoc.fr/item/MMO_2013_74_1_a4/