An arbitrary parallel push-off need not be preferred

By [F1] and [F2], the exterior inclusion and the meridian and longitude classes have the stated meanings. In the basis from [F3], the degree-one property in [F4] writes $[\lambda ]=[\sigma ]+q[\mu ]$ for a unique integer $q$. The curve $\gamma (t)=(e^{2\pi it},e^{2\pi i(q+1)t})$ is embedded because its first coordinate is injective on $S^1$, and it represents $[\sigma ]+(q+1)[\mu ]=[\lambda ]+[\mu ]$. Its projection to the oriented core has degree $+1$.

In the basis $([\mu ],[\lambda ])$ from [F4], the class $[\lambda ]+[\mu ]$ has coordinates $(1,1)$. If it were $md$ for an integer $\mathrm{\mid }m\mathrm{\mid }>1$, both coordinates would be divisible by $m$, which is impossible; hence it is primitive.

The inclusion sends $[\lambda ]+[\mu ]$ to $[\mu ]$ because $[\lambda ]\in \ker i_{\ast }$. The latter is a generator of $H_1(E_K;\mathbb{Z})\cong \mathbb{Z}$, so this image is nonzero. The primitive degree-one class constructed above therefore contradicts the refuted assertion.