Twisted Iwasawa invariants of knots

Abstract

Let $p$ be a prime number and $m$ an integer coprime to $p$. In the spirit of arithmetic topology, we introduce the notions of the twisted Iwasawa invariants $\lambda ,\mu ,\nu$ of ${\mathrm{GL}}_N$-representations and $\mathbb{Z}/m\mathbb{Z}\times {\mathbb{Z}}_p$-covers of knots. We prove among other things that the set of Iwasawa invariants determine the genus and the fiberedness of a knot, yielding their profinite rigidity. Several intuitive examples are attached. We further prove the $\mu =0$ theorem for ${\mathrm{SL}}_2$-representations of twist knot groups and give some remarks.

arXiv comment19 pages, 1 figure; some compile errors are corrected in v2, slight modification in v3, some important corrections in v4