Dual braid monoids, Mikado braids and positivity in Hecke algebras

We study the rational permutation braids, that is the elements of an Artin-Tits group of spherical type which can be written $x^{-1} y$ where $x$ and $y$ are prefixes of the Garside element of the braid monoid. We give a geometric characterization of these braids in type $A_n$ and $B_n$ and then show that in spherical types different from $D_n$ the simple elements of the dual braid monoid (for arbitrary choice of Coxeter element) embedded in the braid group are rational permutation braids (we conjecture this to hold also in type $D_n$).This property implies positivity properties of the polynomials arising in the linear expansion of their images in the Iwahori-Hecke algebra when expressed in the Kazhdan-Lusztig basis. In type $A_n$, it implies positivity properties of their images in the Temperley-Lieb algebra when expressed in the diagram basis.