Visible crew/equipment: When Dr. Van Gelder tells Kirk that he wants asylum and his word that he won't be taken back to Tantalus, in the wideshot from behind Kirk as he moves to his right the two tape marks can be seen on the floor beside Van Gelder's feet.
Plot hole: For a penal colony, Tantalus V has really poor security. Van Gelder manages to escape with ease, by hiding in a box - the only box - beamed up to the Enterprise.
Answer: Kirk was getting his physical and Dr. McCoy probably turned off communications, because if he hadn't, Kirk would have left and headed straight for the bridge, leaving McCoy irritated.