What is required of someone who swore "I swear by Allah" twice that two specific individuals know nothing about a certain matter, and then it became clear to him that he was mistaken and that they do know about it, considering that he gestured with his thumb when he swore?

This oath is not binding, and no expiation is required for it, because it is not concerning a future matter. It appears that you swore an oath believing yourself to be truthful, and then your error became evident, so there is no sin upon you. Oaths are of four types:

1. Binding Oath (Yamin Mun'aqidah): This requires expiation upon breaking it. It pertains to a future matter upon which the heart is set, as Allah the Almighty says: ﴿وَلَكِنْ يُؤَاخِذُكُمْ بِمَا عَقَّدْتُمُ الْأَيْمَانَ﴾ [But He will hold you accountable for what you have earnestly intended in your oaths].

2. Thoughtless Oath (Lagw al-Yamin): This requires no expiation, as Allah the Almighty says: ﴿لَا يُؤَاخِذُكُمُ اللَّهُ بِاللَّغْوِ فِي أَيْمَانِكُمْ﴾ [Allah will not hold you accountable for thoughtlessness in your oaths]. First type: What is uttered unintentionally, such as saying: "No, by Allah," or "Yes, by Allah." Second type: Swearing an oath about something believing it to be true, and then its error becomes evident. According to the apparent stance of the madhhab, no expiation is required for this.

3. Perjury (Yamin al-Ghamus): This is a false oath sworn by someone who knows it is false. According to the apparent stance of the madhhab, it requires no expiation because it is not a binding oath. It has been reported that it is among the major sins for which there is no expiation. Some say expiation is required for it.

4. The fourth type: Not mentioned in the excerpt.