Was Tywin Lannister involved in the murder plot?

Throughout Game of Thrones it's implied that Olena Tyrell and Pyter Baelish were involved in Joffrey's murder and they pulled this off with the help of Sansa and Ser Dontos of House Holard.
But could his grandfather and the Hand of the King be possibly involved with his murder?
Take a look at the fact that Joffrey acted weirdly three times with Tywin Lannister and that made him seriously angry.
As soon as he died he started giving life lessons to Tommen and he was okay with killing children as he gave the order to kill infants of Rheagar and Ellia Martel and just before the Wedding Lady Olena and Tywin had a little chat.
Best Answer
No, Tywin Lannister had no hand in the murder of his grandson Joffrey. Rather than explain how the murder occurred (detailed very well by Paulie_D's answer), I'll analyze Tywin and why he didn't, or couldn't, have done it.
Tywin, at the end of the day, is both a power player and a family man. Almost everything Tywin does is for the sake of either boosting his own power or the name of his family, usually by putting his three kids in places of influence and power. Cersei, as queen. Jaime, as a member of the King's guard, and Tyrion, as Hand of the King (and over various roles over time).
Yes, Joffrey openly defied Tywin on several occasions, but Tywin is the kind of person that plays the long game and will use a person's interests to further his own interests.
After Joffrey's murder, Tywin was already wheeling and dealing to get Tyrion out of the situation as best he could. Tyrion, his most hated son, was still important to him. If Tywin was still willing to work with Tyrion, he surely had enough patience to cultivate Joffrey into, if not a GREAT leader, at least a less murderous one.
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Did Tywin know Tyrion was innocent?
He'll bite his own tongue out before he admits, but Tywin knows Tyrion too clever to poison the king in public so dramatically, that Tyrion is likely innocent, but goddamn he hates Tyrion so much and really wants to believe that Tyrion is in fact the vile, treacherous, disgusting monster Tywin pegged him to be.Was Tywin cruel?
Tywin is severe, ruthless, cruel etc when he needs to be but is also an effective and trustworthy person should he not be in a hostile situation.Did Tywin sleep with Shay?
Having an affair with Tywin was just the cherry on top of the horror that Tyrion faced. Shae's betrayal, of course, ended up setting off a chain of events that ultimately led to Tywin's death in Game of Thrones and Tyrion's separation from House Lannister.Was Tywin Lannister hand of the mad king?
A highly capable administrator, Tywin served as Hand of the King to Aerys II Targaryen for twenty years until the Mad King, driven by envy and distrust, rejected Tywin's proposal to marry his daughter Cersei to Prince Rhaegar and made his son Jaime a member of the Kingsguard.Tywin Lannister History (COMPLETE)
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Answer 2
No the culprit was....
Olenna Redwyne Tyrell, aka the Queen of Thorns
At least, so far...according to George R.R. Martin1
1 - Now confirmed as per Season 7 Episode 3
Martin: In the books — and I make no promises, because I have two more books to write, and I may have more surprises to reveal — the conclusion that the careful reader draws is that Joffrey was killed by the Queen of Thorns, using poison from Sansa’s hair net, so that if anyone actually did think it was poison, then Sansa would be blamed for it. Sansa had certainly good reason for it.
The reason I bring this up is because I think that’s an interesting question of redemption. That’s more like killing Hitler. Does the Queen of Thorns need redemption? Did the Queen of Thorns kill Hitler, or did she murder a 13-year-old boy? Or both? She certainly had good reasons to remove Joffrey. Everything she’d heard about him, he was wildly unstable, and he was about to marry her beloved granddaughter. The Queen of Thorns had studied Joffrey well enough that she knew that at some point he would get bored with Margaery, and Margaery would be maltreated, the same way that Sansa had been. Whereas if she removed him then her granddaughter might still get the crown but without all of the danger. So is that a case where the end justifies the means? I don’t know. That’s what I want the reader or viewer to wrestle with, and to debate.
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