Movie Moves for Knotted Surfaces with Markings
  We present a marked analogue of Carter and Saito's movie theorem. Our definition of marking was chosen to coincide with the markings that arise in link Floer homology. In order to deal with complications arising from certain isotopies, we define three equivalence relations for marked surfaces and work over an equivalence class of marked surfaces when providing our generalization of Carter and Saito's movie theorem. 
Matthew Graham
math.GT