I'm thinking about doing my Master's Thesis on the subject of Arithmetic Topology: the relation between knots and primes, 3-manifolds and number fields, and other connections between topology and number theory (based on the works of B. Mazur, M. Morishita et al.).
I've talked about it with my topology professor and he thinks it's an interesting subject but he does not know if there is enough room in the subject for me to add something new to it.
There seems to be a very small amount of books and articles on the subject and that leads him to believe that there's nothing new and of significance to discover (at least with the mathematical tools that exist today).
I'm not versed enough in the subject to answer that but perhaps one of you can confirm or deny my professor's opinion.