Plot hole: The damage caused by the shuttle crashing into the Bi-Al Foundation is seen BEFORE it actually happens.
Plot hole: The Doctor's clone is free of the virus - so why is Lowe's clone infected? Perhaps they infected him after cloning, in which case, why is Lowe cloned at all?
Plot hole: In the middle of chasing the Doctor, Stor abandons the idea of controlling Gallifrey in favour of destroying it for no readily apparent reason.
Factual error: Titan is depicted as a Moon-like landscape with no atmosphere. In actual fact, Titan has a nitrogen/argon atmosphere denser than Earth's, and a reddish-orange sky. This was detected almost thirty years before "The Invisible Enemy" was made, so somebody slipped up in doing their research.
Other mistake: When Marius injects the cloned Doctor and Leela into the real Doctor, the syringe is full of air, although the later graphic shows them in liquid.
Visible crew/equipment: In episode four, when the Doctor is congratulating himself on blowing up the Swarm, the shadow of a boom mike can be seen.
Plot hole: After the Doctor and Leela's clones fade, the camera pans to show Leela's knife. But since the knife was created in the cloning process, it should have faded out with Leela, the Doctor and their other personal items.
Continuity mistake: In episode three, the readout strip dangling from K-9's mouth shortens between when he shoots the infected crewman and shoot Leela.
Plot hole: Why doesn't Lowe simply shoot Leela the second her back is turned? Moreover, why does he take them to get medical help? It's made clear that the virus has everything it needs on Titan.
Visible crew/equipment: In the first scene of episode two, as the Doctor collapses on the ground, a person can be seen moving in the back of the set.
Revealing mistake: In episode four, the camera wobbles when the Doctor brings the gun to Stael.
Plot hole: Palmerdale and the others hear someone on the stairs. Skinsale opens the door and, despite Reuben being right in front of him, looks downstairs and calls out to the Doctor before turning and suddenly spotting the lumbering keeper.
Revealing mistake: The knife that Leela kills Meeker with is very obviously loosely attached to his clothes. It wobbles back and forth after he falls.
Other mistake: Several shots are repeated during the implosion of the priory.
Continuity mistake: When the Doctor shoots Stor with the De-Mat Gun, the flash starts on Stor's left-hand side, engulfs him, then jumps substantially to the right before whiting out the frame.
Revealing mistake: In episode 3, the section of wall K-9 knocks down is obviously precut, complete with debris on the floor.
Continuity mistake: After shooting the two guards near the beginning of episode three, K-9's nose blaster retracts twice - one in close-up, once in medium shot.
Continuity mistake: When Palmerdale wakes Harker up and asks him if he can use the telegraph, Harker answers him, then asks him to repeat the question, then answers again.
Continuity mistake: Lowe's "frozen" makeup changes between when he emerges from the freezer and when Leela brings him a cup of coffee.
Plot hole: Leela's decision to stay with Andred is very sudden. They barely speak during the serial.
Answer: TARDISes are generally available for properly authorised use on Gallifrey; they're not usually assigned to a particular Timelord on a long-term basis. The Doctor stole his when he left his homeworld.
Tailkinker ★