Is there any connection between knot theory and number theory in any aspects? Does anybody know any book that is about knot theory and number theory?
The question seems very general, but the first book to come to mind is this:
The Arithmetic of Hyperbolic 3-Manifolds, with C. Maclachlan, Graduate Text in Math. 219, Springer-Verlag (2003)
Currently, the most complete book presenting the connections between knot theory and number theory is "Knots and Primes. An Introduction to Arithmetic Topology" by Masanori Morishita (Springer, 2012). (It also presents much of the needed prerequisites, so it should be the first choice for somebody serious about self-studying the subject.)
If you feel confident enough that you already know the basics, it might be useful to take a look at "Primes and Knots" edited by Toshitake Kohno and Masanori Morishita (AMS, 2006). It contains the proceedings of two 2003 conferences on this topic and closely related ones (and, strangely, it is currently freely available, so grab it legally while you can!).
I am not at all an expert in these matters, but there seems to be recent developments in this area around the work of Minhyong Kim and collaborators. See in particular:
Since the classic article by Witten, Chern-Simons gauge theory from physics is well known to be related to knot theory. My understanding is this work by Kim and collaborators aims at developing an arithmetic analogue of this theory.
I recommended look lower link.