How much do I have to know about a group's research before I email the PI about openings? My field of study is purely theoretical, meaning that there are a ton of equations and a bunch of numerical simulations. I am targeting groups that align with my field and I am trying to read their research. I know this question has probably been asked before, but how much of the group's research am I expected to know before I email the PI about lab openings? Three levels of abstraction are given below:

Just the title of the projects and the material given on the webpages.
Go through the abstract and conclusions of their latest papers.
Take a single paper and go through it line by line, understanding all the equations and simulations.
For me, it really depends on the stage of interaction with our group.

For the first contact, I would expect that you have at least visited our website and know what we work on, in general. Browsing the website and some paper titles should be fine for this. If none of the paper titles make sense or sound interesting to you, then why are you applying to work with us in the first place?
If we end up chatting more detailedly (e.g., if you have a phone interview with members of my lab) I would expect that you looked at all the recent papers involving the people interviewing you, and have read at least a subset of those. I don't care whether you understand the work line-by-line, and I am fully aware that some of our work can be technical and academic. However, if you can't be bothered to at least skim some of our papers and try to grasp the core ideas, it does not reflect very well on your commitment to our line of research.