The following theorem, which can be deduced from a combination theorem of Dahmani, is stated as Theorem 7.2.2 in our book *3-manifold groups* .

**Theorem:** Let $N$ be a compact, orientable, irreducible 3-manifold with empty or toroidal boundary. Let $M_1,\ldots,M_k$ be the maximal graph-manifold pieces of the JSJ decomposition of $N$, let $S_1,\ldots,S_l$ be the tori in the boundary of $N$ that adjoin a hyperbolic piece and let $T_1,\ldots,T_m$ be the tori in the JSJ decomposition of $N$ that separate two (not necessarily distinct) hyperbolic pieces of the JSJ decomposition. The fundamental group of $N$ is hyperbolic relative to the set of parabolic subgroups
$$
\{H_i\}=\{\pi_1M_p\}\cup\{\pi_1 S_q\}\cup\{\pi_1 T_r\}~.
$$

In particular, such a 3-manifold $M$ is non-elementarily relatively hyperbolic if and only if it is not a graph manifold (including the possibility of being Seifert fibred or a Sol manifold).

I believe that, historically, knots whose complements are graph manifolds were called *hose knots*, so the statement is that a knot complement group is non-elementarily relatively hyperbolic if and only if it's not a hose knot.

You also ask about *acylindrical* hyperbolicity, a weaker notion which merely asks for some acylindrical action on a hyperbolic space.

It turns out that non-geometric graph manifold groups always act acylindrically on the Bass-Serre tree of the JSJ decomposition (see Lemma 2.4 here), and so are acylindrically hyperbolic. So the fundamental group of a knot complement will be acylindrically hyperbolic unless the knot is Seifert fibred.

Finally, the remaining Seifert fibred knot complements (except for the trivial knot) all admit $\mathbb{H}^2\times\mathbb{R}$ geometry. In particular, after quotienting out a central infinite-cyclic subgroup, they act properly on the hyperbolic plane. The central $\mathbb{Z}$ acting trivially is the reason the corresponding action of the knot group on the hyperbolic plane fails to be acylindrical -- so even in this case, the groups are "nearly" acylindrically hyperbolic.