The Ouroboros is an ancient Egyptian symbol depicting a serpent forming a circle as it devours its own tail. The symbol represents eternal repetition, and the infinite cycle of life, death and rebirth.
Daniel hides his black eye under a pair of dark sunglasses before going to work. They are the exact same style of aviators that Daniel wore to school to cover his black eye after his first encounter with Johnny Lawrence.
When Daniel goes in Miyagi's room, he sees the note Miyagi left for him about painting the house. It had been folded but now lays open on the table. However, in the movie Daniel wadded up the note and threw it on the ground. The note in the show has clearly not been wadded up.
The title is "Ouroboros" but the symbol is never shown.