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?
5$\begingroup$ Welcome new contributor. Perhaps you are looking for Mazur's dictionary: neverendingbooks.org/mazurs-dictionary $\endgroup$– Jason StarrMar 19, 2019 at 13:15
6$\begingroup$ The answers so far are about the analogy between primes and knots, which typically takes inspiration from knot theory to try to do some number theory. There’s some small amount of work going the other direction, notably by Alison Miller (math.harvard.edu/~abmiller). $\endgroup$– Daniel LittMar 19, 2019 at 13:59
$\begingroup$ @DanielLitt thank you for promoting research outside the mainstream understanding. $\endgroup$– Matt CuffaroMar 19, 2019 at 14:37
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)
I recommend Chao Li/Charmaine Sia's notes for a brisk and illustrated overview of the ever-growing MKR dictionary detailed in Knots and Primes. The first three or so chapters of the source book is rather accessible.
For the MKR dictionary, refer to the link in comments.
1$\begingroup$ An alternative link for Li and Sia's notes, in case the above dies, is under Li's page at Columbia University. $\endgroup$– Alex M.Mar 21, 2019 at 18:03
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.