When it comes to the relationship between Claudius and Hamlet, that is.
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The obvious answer is that Claudius just could have admitted his sin and all of the deaths that came after the death of Hamlet Sr. could have been avoided. Claudius becomes more and more suspicious of Hamlet and there is confusion since only he and Hamlet (and Horatio) know about the secret murder. Claudius arranges with Laertes to kill Hamlet in the "duel" at the end using poison on the tip of the sword and in the poison cup; ultimately his plan to kill Hamlet results in the deaths of Gertrude, Laertes, Hamlet, and Claudius himself. The friends who took Hamlet to England to have Hamlet killed by the English king results in the deaths of Rosencrantz and Guildenstern. The conflict between the two is the center of all the murders.