What evidence is there that Gertrude was innocent of complicity in poisoning her King Hamlet in Act III, Scene 4 of Hamlet?
Initially, Gertrude hopes to discover the cause of Hamlet's bizarre behavior as Polonius eavesdrops behind a tapestry in her room. When Hamlet enters Gertrude's chamber, he accosts his mother, and she responds by challenging her son's negative views of her. Hamlet then hears a noise from behind the tapestry and unknowingly murders Polonius, thinking that Claudius is behind the tapestry. Following Polonius's murder, Gertrude says that Hamlet has conducted such a senseless, horrible act. Hamlet responds by saying,
A bloody deed? Almost as bad, good mother, As kill a king and marry with his brother. (Shakespeare, 3.4.29-30)
Gertrude reveals her innocence by asking Hamlet,
As kill a king? (Shakespeare, 3.4.31)
She is clearly unaware of what Hamlet is talking about and does not immediately recognize that Hamlet is referring to how Claudius murdered his brother. If Gertrude were complicit in her husband's assassination, she would not have questioned Hamlet and fully understood why he is upset with her.
As Act Three, Scene 4 continues, Hamlet accosts his mother for marrying Claudius and getting over the death of his father so quickly. Gertrude feels terrible about her decision to immediately marry Claudius and expresses her guilt. However, Gertrude never states that she feels bad for colluding with Claudius or participating in King Hamlet's murder. Overall, Gertrude is portrayed as a naive, heart-broken woman in this scene and there is no evidence that she was complicit in her husband's murder.
When Hamlet is engaged in a discussion with Gertrude, they are sparring with words. She is talking about one thing, he another. She tells him he has offended his father very much (meaning Claudius, who is not his father), and he tells her that, no, madame, it is you who have offended my father - meaning by marrying Claudius. He then proceeds to say that it is an offense to have murdered a king, and she does not know what he is talking about - "Is the king dead?" she asks. She seems to be totally unaware that Hamlet is insinuating that Claudius has killed his father, and Gertrude's former husband. Hamlet has not been sure whether or not his mother knows about the murder, and in this scene, it indicates that although she is a huge air-head, she really did not know about Claudius' misdeeds. We are never really sure whether Gertrude has some knowledge of what happened, whether she is mentally unbalanced and is in denial, or she is just a huge airhead (this is my view). She truly seems not to know that her first husband has been poisoned by her current husband. Also, she does not know why Hamlet is so upset and if she knew about the poisoning, she would understand his strange behavior.
The purpose of this conversation between Gertrude and Hamlet is for Gertrude to try and determine what is going on with Hamlet. She starts out on the offensive, accusing him of offending his (step-) father, Claudius. Hamlet is already in high spirits having just missed an opportunity to kill Claudius before he enters his mother's rooms, and he is ready to say everything and anything that comes to mind. He even claims he wishes she were not his mother. When he mistakenly kills Polonius the Queen exclaims, "What a rash and bloody deed is this!" Hamlet seizes this opportunity to put out the idea of his father being killed and then watches Gertrude's reaction to the statement. By her reaction, he may be able to judge whether she has knowledge of Claudius' crime. He says, "Almost as bad...As kill a king, and marry with his brother." Gertrude's immediate response is, "As kill a king?" What she means is, "What are you talking about?" Because Hamlet doesn't pursue this topic further, we can conclude that he judges his mother to be innocent of the murder of King Hamlet. He proceeds to try and convince his mother to leave Claudius; he doesn't even hint to her about his knowledge of Claudius' actions.