Title : An analogue of non-commutative Iwasawa theory for fibered knots

Abstract : It is known that there are deep analogies between low-dimensional topology and algebraic number theory. In particular, Alexander-Fox theory is considered to be the topological counterpart of Iwasawa theory. In this talk, based on the insights from non-commutative Iwasawa theory, we construct a knot invariant for fibered knots, which is a refinement of classical Alexander polynomials. As an application, we generalize Fox’s classical formula for the orders of first homology groups, extending it from cyclic covers to certain non-abelian covers of $S^3$ branched along knots. We further explore several consequences of this formula.