A LEGENDRIAN STRATIFICATION OF RATIONAL TANGLES

Abstract

A subset of legendrian 2-string tangles are defined to be minimal if the strands realize the minimum absolute value of the Bennequin invariant. The restrictiveness of this condition is examined by studying which topological rational tangles have minimal representatives. It is shown that a rational of finite parity has a minimal representative if and only if its standard continued fraction expansion contains only non-negative entries, is of odd length, and has every horizontal entry even. This is proved by applying recent results of Fuchs and Tabachnikov or Chmutov and Goryunov that give an upperbound for the Bennequin invariant of links in terms of the minimal exponent of the framing variable of the Kauffman polynomial and Yokota's precise formula for this topological invariant. A second geometric stratum of rational tangles with infinite parity is defined and it is then shown a positive rational has a minimal represenative if and only if its continued fraction expansion is a particular extension of a minimal of finite parity and a negative rational of infinite parity has a minimal representative if and only if it has even length with every vertical entry even.