Movie moves for knotted surfaces with markings
We present a marked analog 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 isotopes, we define three equivalences for marked surfaces and work over an equivalence class of marked surfaces when proving our generalization of Carter and Saito's movie theorem.
Journal of Knot Theory and its Ramifications
Graham, Matthew, "Movie moves for knotted surfaces with markings" (2018). Mathematics Faculty Publications. 7.