When Fry is showing the note to the professor, it is missing the tear at the top. When the professor is putting the note into the analyzer, the tear is back.
Bender referred to Leela as a mutant before they all found out she was a mutant.
None of the mutants have any problem getting to the surface, but Leela, Bender, and Fry need to take a hot air balloon? Also, every time Bender drops stuff on Leela and/or Fry, he's within 20 feet of them. So, why the balloon?