Sans is a weird one. I think, even despite their being direct co-workers and both becoming supervisors around the same time, it probably took Amethyst rooming with Sans (and thus circumstances forcing them to hang out) for them to really become friends. Sans can be so cryptic, which can be a struggle for someone who often has trouble navigating casual conversation like Peridot does… That, and his lazy habits and what she PERCEIVES as a flippant attitude (read: telling jokes at INAPPROPRIATE TIMES) initially gave Peridot the impression that they had very little in common.

Buuuuut it’s kind of hard to keep disliking a guy as affable as Sans is when you’re exposed to him as constantly as Peridot has been. There’s no major tide-turning moment that she can point to and be like “yup, right there, that’s when we became friends”. It just kind of happened. Now he’s on that shortlist of people who she will put herself in harm's way for. Interfering in that fight in the courtyard was just as much about Sans as it was Lambert, and hearing the Prince refer to Sans as “his” servant really struck a nerve. Because he’s not! He’s the Ringmaster’s warden, and Peridot’s buddy. And she wants him back. I’m pretty sure she wants him to help her with some kind of big wacky invention in the future and she’s not sure how he’s going to do that if he’s all brain washed.