$\newcommand{\ab}{\mathrm{ab}}$Let $L=K_1\cup \dots \cup K_r$ be a link embedded in a 3-sphere. Here, $K_1,\dots, K_r$ are the component knots of $L$. A prototypical invariant associated with $L$ is its Alexander polynomial. This is a polynomial in $\Lambda_r=\mathbb{Z}[T_1^{\pm 1}, \dots, T_r^{\pm 1}]$, the Laurent series ring in $r$-variables. It is defined as follows. Let $X_L$ be the complement of $L$ in $S^3$ and $\widetilde{X}_L^{\ab}\rightarrow X_L$ be its universal abelian cover. The Deck group of this cover is the abelianization of $\pi_1(X_L)$, which by the Hurewicz theorem is isomorphic to $H_1(X_L;\mathbb{Z})$. This homology group is generated by $x_1,\dots, x_r$, where $x_i$ is the class associated with the meridian looping around that $i$-th knot $K_i$. The Alexander module is $H_1(\widetilde{X}_L^{\ab}, \mathcal{F}_{x_0};\mathbb{Z})$, where $x_0$ is a chosen base-point of $X_L$ (where all meridians begin and end at), and $\mathcal{F}_{x_0}$ its fibre in $\widetilde{X}_L^{\ab}$. This becomes a module over $\pi_1(X_L)^{\ab}$ and hence, is a module over $\Lambda_r$ as well. The multivariable Alexander polynomial is then defined to be the 1-st elementary ideal of the Alexander module. An account of this can be found in any basic text in knot theory.

Fix a prime $p$. There is a p-adic version of this construction due to Hillman, Matei, and Morishita, an account of which can be found in the book "Knots and primes" by Morishita. The p-adic analog of the Alexander polynomial is defined in the power series ring $\widehat{\Lambda}_r:=\mathbb{Z}_p[\![X_1,\dots, X_r]\!]$ and is the topological equivalent of the Iwasawa polynomial defined for number fields. In fact, there is an asymptotic formula for the growth of $p$-parts of homology classes in various $\mathbb{Z}/p^n\mathbb{Z}$-covers that can be determined in terms of Iwasawa invariants $\mu,\lambda$ and $\nu$, once a $\mathbb{Z}_p$-tower is fixed. For more information, see the last few chapters in Morishita's book, or the paper of Hillman-Matei-Morishita.

Given these developments, it is natural to extend the analogy between number theory and topology further. In Iwasawa theory, it is natural to work with non-commutative towers and extensions and study non-commutative versions of Iwasawa invariants and polynomials. This has been done for number fields. There is a nice book on the subject called "Noncommutative Main conjectures over totally real fields" for instance. Perhaps one can study non-commutative analogs of the Alexander polynomial which will have significant knot theoretic applications as well, and perhaps one can compute these in some examples. After all Alexander polynomials can be computed using a variety of techniques, like using Seifert surfaces. I wonder how difficult it would be to tackle such questions since, after all, I'm not a knot theorist. But I don't see anything in the literature that suggests this problem has been studied.