Given a (tame) knot $K \subset S^3$, let $t \in G = \pi\_1(S^3 - K)$ be any meridian. The Wirtinger presentation shows that $\langle \langle t \rangle \rangle = G$, where the notation indicates the normal closure of $t$ in $G$.


Let $I$ be an infinite subset of the natural numbers. A linking number argument shows that
$$
\bigcap\_{i \in I} \langle \langle t^i \rangle \rangle \subset [G,G].
$$
My question is: when is this intersection trivial? When does
$$
\bigcap\_{i \in I} \langle \langle t^i \rangle \rangle = \{1\}?
$$
In particular, does this hold if $I$ is the set of all powers of $2$? Is there a known example where the above does not hold?



Powers of meridians in knot groups
For hyperbolic knots, this holds for any infinite set $I\subset \mathbb{N}$ via hyperbolic Dehn filling. One may use Thurston’s original theorem, or the geometric group theory version of [Groves-Manning](https://mathscinet.ams.org/mathscinet-getitem?mr=2448064) and [Osin](https://mathscinet.ams.org/mathscinet-getitem?mr=2270456). I will describe a version using the Gromov-Thurston $2\pi$ theorem since this is how I think about it intuitively. To understand this argument you need to know a bit about Riemannian geometry and hyperbolic [orbifolds](https://en.wikipedia.org/wiki/Orbifold?wprov=sfti1) as well as consult some of the linked references.


Consider a tame knot $K\subset S^3$ with hyperbolic metric $\rho$ on $M\_K=S^3-K$. Let $x\in \pi\_1(M\_K)-\{1\}$. There is a peripheral torus $T=\partial (\mathcal{N}(K))\subset M\_K$ with $\pi\_1$-injective fundamental group, such that $\pi\_1(T)\subset \pi\_1(M\_K)$ is generated by elements $\mu$ the meridian, realized by a little loop in $T$ linking $K$, and $\lambda$ the longitude which is trivial in $\pi\_1(T)\to H\_1(T)\to H\_1(M\_K)$. If $x$ is not peripheral, then it is realized by a closed geodesic $\gamma:S^1\to M\_K$ immersed in $M\_K$, meaning that $x=\gamma\_\#(s)$, where $s\in \pi\_1(S^1)$ is a generator (I am suppressing basepoints in the notation).


In the metric $\rho$ on $H\_K$, there is a tubular neighborhood $C\subset M\_K$ called a horocusp which is the image of a horoball in the universal cover, such that $\partial C$ is isotopic to $T$ in $M\_K$. We may shrink $C$ down to a smaller horocusp $C\_x$ such that $C\_x \cap \gamma(S^1)= \emptyset$. Choose a number $n\in I$ such that $length([n\mu]) > 2\pi$, where $[n\mu]$ is a geodesic curve in $\partial C\_x$ realizing the homology class $n\mu\in H\_1(C\_x)=H\_1(T)$. Then the [Gromov-Thurston $2\pi$ theorem](https://mathscinet.ams.org/mathscinet-getitem?mr=1396779) implies that one may fill in an orbifold negatively curved cone metric with cone angle $2\pi/n$ along the meridian (this manifold has the notation $M\_K(n,0)$ in [SnapPea/SnapPy](https://en.wikipedia.org/wiki/SnapPea?wprov=sfti1)). The orbifold fundamental group is $\pi\_1(M\_K)/<<\mu^n>>$, the group obtained by killing the $n$th power of the meridian. Then the element $x\in \pi\_1(M\_K)$ will be non-trivial in $\pi\_1(M\_K(n,0))$, since it is realized by a closed geodesic.
The case that $x$ is peripheral I will leave as an exercise (hint: the subgroup $<\lambda, \mu>/<<n\mu>>$ will inject into $\pi\_1(M\_K(n,0))$).


I think this proof should generalize to arbitrary knots, but I haven’t thought it through carefully.


