I am looking for references on the geometry of knot groups. For instance, I am interested in the following question:



> 
> When is a knot group relatively hyperbolic?
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Hyperbolic knots are known to have a group which is relatively hyperbolic, but is the reciprocal true? Is the answer easier when restricted to alternating knots? And what about acylindrically hyperbolic knot groups?



Relatively hyperbolic knot groups
The following theorem, which can be deduced from a [combination theorem](https://arxiv.org/abs/math/0203258) of Dahmani, is stated as Theorem 7.2.2 in our book *[3-manifold groups](http://www.ems-ph.org/books/book.php?proj_nr=195)* .



> 
> **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
>  $$
> \{H\_i\}=\{\pi\_1M\_p\}\cup\{\pi\_1 S\_q\}\cup\{\pi\_1 T\_r\}~.
> $$
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> 
> 


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](https://arxiv.org/abs/0807.3727)), 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.


