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.Comment: 50 pages, 62 Figures, numerous minor correction