Research Seminars @ Illinois

Arithmetic topology, also known as the "knots and primes" analogy, studies connections and analogies between low dimensional topology and number theory. In low dimensional topology, McMullen constructed a sequence of knots linked in a "Chebotarev way", and it may be seen as an analogue of the set of all prime numbers. In my talk, I will outline the above and discuss an ongoing joint work with Jun Ueki and Yi Wang, in which we use this notion to provide a topological version of Nuekirch-Uchida theorem. In other words, we construct a profinite group of a 3-manifold relative to the link, which is the "absolute Galois group" of the manifold. We show that such a group determines the manifold. In our arguments we study analogies of fundamental number theoretic concepts such as Hilbert ramification theory and L-functions for links.