From the film "Taxi Driver" (1976). The central character (played by Robert De Niro) goes doolally and society is to blame. During his journey into madness and mayhem, he decides to shave his hair into a mohican style or mohawk. This hairstyle has since been particularly embraced by punk rockers thanks mainly to this movie and the bands Plasmatics and Discharge. The De Niro character is called Travis Bickle... therefore a "Travis Cut" ("Bickle Cut" sounds too much like a chunky sandwich spread).
Travis Cut were also the name of a UK punk band from the mid 1990's.