I'm currently reading Freedman's paper on the Mobius energy of knots. In the proof of Theorem 4.3, he constructs a cylindrical covering $N$ of a tame knot $\gamma_K$, contained a tubular neighborhood $M$ of said knot. He claims that $\gamma_K$ is a core curve of the cylindrical covering, and it appears it's done through an entirely algebraic argument.
Specifically, he claims that by the Van Kampen theorem, $\pi_1(M - \gamma_K) = \pi_1(N - \gamma_K) \ast_{\pi_1(\partial N)} \pi_1(M - \text{int}(N))$. As $\pi_1(M-\gamma_K) = \pi_1(\partial N) = \mathbb{Z} \oplus \mathbb{Z}$, we can deduce both summands of the free product with amalgamation are also $\mathbb{Z}\oplus \mathbb{Z}$. Is there an algebraic reason why this must hold? In other words, if $G \ast_K H = \mathbb{Z} \oplus \mathbb{Z}$, and $K = \mathbb{Z} \oplus \mathbb{Z}$, can we deduce that $G = H = \mathbb{Z} \oplus \mathbb{Z}$?
No: let $G= \mathbb{Z} * \mathbb{Z} = F(x,y)$ and $H= \mathbb{Z} =F(z)$, and write $K= \mathbb{Z} \oplus \mathbb{Z} = F(a,b)/(ab=ba) $. Define $\varphi_1: K \to G$ by $\varphi_1(a) := xyx^{-1}y^{-1}$ and $\varphi_1(b):=1$ (note it is well-defined in the quotient) and $\varphi_1: K \to H$ as $\varphi_2(a) :=1$ and $\varphi_2(b) :=z$. Then $$G *_K H = F(x,y,z)/(xyx^{-1}y^{-1}=1, z=1) = F(x,y)/(xy=yx) = \mathbb{Z} \oplus \mathbb{Z}.$$