Slits, gills, rips, etc.

I have a surface.  I want to make a bunch of lines, use the lines to cut "slits" in the surface, and pull the control points up.  I basically need to rip the nurb at the isocurves, and pull up the control points. 

I have already figured out the replace and pull thing.  I just need to figure out the "rip" thing.