Amelapidot

Amelapidot is the poly ship between Amethyst, Lapis Lazuli and Peridot from the Steven Universe fandom.

Canon
Peridot is great friends with both Lapis and Amethyst. She lives at the barn with Lapis, and occasionally hangs out with Amethyst as well.

Peridot
At first Lapis was hesitant to live with Peridot. Peridot tried to convince her by giving her gifts, a card, a pool, and her favorite item, the tape recorder. Still, even after that Lapis rejects her. After Peridot talks to her and says that she understands how she feels, she is more willing to be friends with her. Lapis enjoys Peridot's company. They like to make art (or as they call it, "meep morp") and music together, as well as watch "Camp Pining Hearts". When Peridot reveals that she's been moderating her speech when around Lapis to avoid upsetting her, Lapis got angry and left, taking the barn with her into space, leaving Peridot alone. After some time though, she returns. As of "Steven Universe: The Movie" they seem to be on good terms.

Amethyst
The two have not interacted much, but they are presumably on good terms, as they are both Crystal Gems now.

Lapis
Peridot seems to admire Lapis, as she was the one who was more willing to live on the barn with her, even trying to give her gifts to convince her. She gives her gifts, and after the attempts to make it up to her failed, she talked to Lapis and explained that she understands how she feels, they agree to live together. She is happy that Lapis cares about her. She seemed deeply upset when Lapis left with the barn, and after she came back, she was very surprised, running to her and asking if she's really back. As of "Change Your Mind" and "Steven Universe: The Movie" they seem to be getting along well, hanging alongside Bismuth and building Little Homeworld.

Amethyst
Peridot sees Amethyst as "above" the other Crystal Gems, as she is a quartz, the only "real" gem. She is shown to care about her, saving her from the drill that was about to hit her, pushing her out of the way. She says the beta kindergarten is not nearly as impressive as the kindergarten Amethyst came out of. She has also said that she finds her company entertaining. When Amethyst gave her nicknames, Peridot learned from her and gave her some back (Ams, Big A, etc.). She gets excited when she sees her, yelling "Amethyst!". She tries to impress her with her "meep morps" and music. She also thinks that Amethyst is way better than Jasper, despite Jasper being non-defective. In "Change Your Mind", Peridot is shown gleefully running up to and climbing on top of Amethyst after finding out about the latter's reformation. This shows that the two still appear to be good friends.

Peridot
Amethyst cares about Peridot, telling her she likes her for who she is, not who she could be. She likes Peridot's jokes, but sometimes she finds them hurtful. After finding out how Amethyst feels, Peridot apologizes to her in a recorded message. Amethyst still sees Peridot as a nerd, and likes to call her one. She enjoys giving Peridot nicknames, such as "P-Dot" or "Peridactyl". In "Too Short To Ride" she was determined to get Peridot the Alien toy she wanted. In "Back To The Kindergarten" she attempts to cheer her up by planting sunflowers in it.

Lapis
Though the two have not interacted a lot, they seem to be on good terms, as they are both Crystal Gems.

Fanon
FAN FICTION

Amethyst/Lapis Lazuli/Peridot tag on AO3

TUMBLR

lapamedot tag on Tumblr

amelapidot tag on Tumblr

lamedot tag on Tumblr

Fandom
It is not very popular in the fandom, having only 48 fanfiction works on AO3, and not having many posts on Tumblr.

Trivia

 * This ship was presumably made to end the wars between Amedot and Lapidot, though its shippers also ship it because they like both ships equally and think it's cute.
 * Peridot has interacted with both Amethyst and Lapis and is friends with both of them, however Amethyst and Lapis have not interacted much with each other.