This is the first new episode of Columbo in a decade. Peter Falk was 61 when he returned to the role.
On the wall of Max's workshop is a framed poster of Randi - this is The Amazing Randi (James Randi), a magician turned skeptic and debunker of fake psychics, who is the model for the character of Max Dyson.
Columbo states he's never seen anything like the guillotine trick, but this isn't true. When he dealt with another magician in Now You See Him (1976), he goes to a magic shop where the clerk demonstrates a smaller version of this very trick, first using a carrot and then using Columbo's hand.
The psychic research foundation where the testing takes place is the Annemann Institute. This is an homage to Theodore "Theo" Annemann, a famous American magician from the first half of the 20th century. His specialty was Mentalism and he heavily influenced both mentalists and magicians who came after.