Thanks for the response. I sent V an email. I have more questions, but I think these are easier.
So I think K is a really good match for me, and I want to try my very best to be accepted to his university and subsequently work with him. Many people here suggest emailing POIs for an increased chance of success, so that is what I plan to do.
But the question is: when? I haven't yet submitted my application (the deadline is Jan 15th). Is it better to contact him before I submit my application or after? I'm fairly confident he will take new student(s), but I'm not totally sure. And I'm applying to the school anyway because I have other professors in mind.
Thanks again.