Quantum one-cocycles for tangles and geometry of braids

We construct the first quantum one-cocycles for tangles. There values live in a refined graded HOMFLYPT skein module of five variables. As a first application we show that there are already two-component string links, which are not isotopic to any partially closed braid. As another application we show that our one-cocycles can detect wether a braid is pseudo-Anosov. We formulate an analogue of Kashaevs volume conjecture: the stretching factor of a pseudo-Anosov braid is the limit under iteration of the braid of a certain value of our one-cocycle applied to the rotation of the braid.