Why does Oedipus investigate Laius’s murder?

Oedipus investigates because Apollo has linked the plague to unpunished blood — Thebes will not heal until Laius’s killer is found and driven out. As king, Oedipus accepts that duty publicly. He summons Teiresias, questions Creon, and pursues witnesses with the same force he once brought to the Sphinx’s riddle.

Ironically, the investigation is also his fate completing itself. Each clue — the survivor’s story of a crossroads, Jocasta’s mention of the exposed child, the Corinthian messenger’s news that Polybus was not Oedipus’s father — narrows the circle. He cannot stop without failing as king; continuing destroys him as man. Answers that only say “he is curious” miss the religious-political weight: purification of the city requires truth, and truth is the one thing that will exile him.