I have a question about a subgroup of the free group on three generators, inspired by the following riddle:

Can you hang a painting using a string and two nails so that if either of the nails is removed, the painting falls? [This has been mentioned on the stack before: see this post for a solution and discussion.]

In short, this question is equivalent to asking if there is an element of the free group on $a$ and $b$ whose image under either quotient map $a \mapsto 1$ or $b \mapsto 1$ yields the identity element. (I'm thinking of the free group on two generators as the fundamental group of the plane minus two points; each generator corresponds to wrapping the string around one of the nails.) The commutator $[a,b] = aba^{-1}b^{-1}$ is the simplest solution: the set of elements that work is (I think) exactly the commutator subgroup of $\mathbb{Z} * \mathbb{Z}$.

I want to know about the analogous question for the free group on $a,b$ and $c$: if $f_a, f_b$ and $f_c$ denote the quotients by the generators $a,b$ and $c$, respectively, then what is the intersection $H$ of the kernels of $f_a, f_b$ and $f_c$?

$H$ is a normal subgroup of $\mathbb{Z} * \mathbb{Z} * \mathbb{Z}$, since the intersection of normal subgroups is normal, and it's non-trivial: one element that works is $[a,b]c[a,b]^{-1}c^{-1}$. Is there a nice characterization of this subgroup as in the case with two generators?

This paper has a lot of cool results about this type of question, though their work is mostly about finding the shortest length word that satisfies the condition I'm talking about. As far as I can see, they don't discuss a characterization of *all* solutions to the painting puzzles.

If we can find a solution for the free group on three generators, maybe we can generalize: let $H_k^n$ be the intersection of the kernels of all the quotient maps of the free group on $n$ generators by any distinct $k$ generators. (So $H_1^3$ is what I asked about above.) Is there a simple characterization of the elements in $H_k^n$, similar to $H_1^2$ being the commutator subgroup of $\mathbb{Z} * \mathbb{Z}$?