Reading the dual Garside length of braids from homological and quantum   representations
  We show that Lawrence's representation and linear representations from quantum sl_2 called generic highest weight vectors detect the dual Garside length of braids in a simple and natural way. That is, by expressing a representation as a matrix over a Laurent polynomial ring using certain natural basis, the span of the variable is equal to the constant multiples of the dual Garside length. 
Tetsuya Ito
math.GR math.QA math.RT