Twisted Iwasawa invariants of knots

TL;DR

This paper introduces twisted Iwasawa invariants for knot covers, demonstrating they determine key knot properties like genus and fiberedness, and proves the $mbda=0$ theorem for certain knot groups.

Contribution

It defines twisted Iwasawa invariants for knot covers and establishes their role in determining knot properties, extending arithmetic topology concepts.

Findings

Invariants determine genus and fiberedness of knots.

Proves $mbda=0$ theorem for ${ m SL}_2$-representations of twist knot groups.

Provides examples illustrating the invariants and their applications.

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 ${\rm GL}_N$-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 ${\rm SL}_2$-representations of twist knot groups and give some remarks.