The Hanged Man Arcana
You can start this Social Link any time after July 10th by visiting Naganaki Shrine and speaking to the little girl there, Maiko. You'll need to bring Mad Bull and Weird Takoyaki with you to the shrine, having them in your inventory is a prerequisite to starting this Social Link. You can get the Mad Bull from any vending machine, the one in your Dorm works and the Weird Takoyaki you can get from Tartarus or Octopia which is a store at Iwatodai Station.
Once you speak to Maiko she'll ask you for these two items (as seen in the screenshots below) after you feed her and give her something to drink you'll unlock this Social Link. Maiko is available at the Naganaki Shrine playground Monday - Saturday; she's never here on holidays.
While raising a Social Link you'll be given story cutscenes where you'll have to answer questions. The answers to each of these questions is different depending on the person and the situation. What you usually need to try and do is pick an answer that reflects their personality. Here's a cheat sheet for each Social Link rank of the Hanged Man Arcana and what to say.
Rank 1: Automatically earned
Rank 2: Sure, let's go - Anything - Anything
Rank 3: Anything - Don't worry, he'll be there
Rank 4: I'm happy for you - Nobody could forget you
Rank 5: That's terrible - That isn't true
Rank 6: Calm Down - Sounds Like a Plan - You'll need more than that
Rank 7: Anything - Anything
Rank 8: Hamburgers - You did good - Choose your father
Rank 9: We're friends forever!
Rank 10: Automatically earned
Upon completing this Social Link you'll be able to fuse Attis through Pentagon fusion. You'll need the Personas Ubelluris, Vasuki, Orthrus, Take-minakata and Inugami in order to perform this fusion.