Is $\langle a, b \mid baba^{-1}\rangle$ truly isomorphic to $\langle a, c \mid a^2c^2\rangle$?
Showing that $\left\langle a,b \mid abab^{-1}\right\rangle \cong \pi_1(K) \cong \left\langle c,d \mid c^2 d^2 \right\rangle$