To 'alligate' is to have a photo taken of you with any object covering the bottom half of your face i.e. directly below the eyes. You can use bar menu's, statues, people, buildings or any manner of objects; the possibilities are endless.
**In a drunken haze**
Tom: "Holy shit
look at that statue! Shall we do some alligating on it?"
Lennie "Yeh, lets do it"