Bypasses for rectangular diagrams. Proof of Jones' conjecture and   related questions
  In the present paper a criteria for a rectangular diagram to admit a simplification is given in terms of Legendrian knots. It is shown that there are two types of simplifications which are mutually independent in a sense. A new proof of the monotonic simplification theorem for the unknot is given. It is shown that a minimal rectangular diagram maximizes the Thurston--Bennequin number for the corresponding Legendrian links. Jones' conjecture about the invariance of the algebraic number of intersections of a minimal braid representing a fixed link type is proved. 
Ivan Dynnikov and Maxim Prasolov
math.GT