# Dehn surgery along primitive knot in 3-dimensional handlebody

I'm studying the article "An alternative proof of Lickorish–Wallace theorem" (doi link) and I got stuck in a problem.

Let $$H_g$$ be a 3 dimensional handlebody of genus $$g$$, a primate curve in $$H_g$$ is a knot in $$\partial H_g$$ that intersects an essential disk of $$H_g$$ in a single point. Let $$c$$ be a primitive curve, pushing $$c$$ in the interior of $$H_g$$ we obtain the knot $$c'$$. Now consider a spanning annulus $$A$$ in $$H_g \setminus \eta(c')$$ with $$c \subset \partial A$$, and the other boundary component of $$A$$ is called $$c''$$ and lies in $$\partial \eta(c')$$. How can I prove that if I perform a surgery on $$c'$$ along $$c''$$ I obtain a genus $$g$$ handlebody?

According to my notations, a surgery on $$c'$$ along $$c''$$ means glueing the meridian $$\{x\} \times \partial D^2 \subset S^1 \times D^2$$ on $$c''$$.

I found a similar question (Dehn surgery on handlebody), the answers (in particular the one by Ian Agol) seems to confirm that my statement is true, but there are no details.

Since $$c\subset H_g$$ intersects an essential disc $$D$$ in a single point, the boundary of a regular neighbourhood of $$D\cup c$$ is another disc $$D'$$, which splits $$H_g$$ into a solid torus containing $$D\cup c$$ and the rest. You can forget about the rest (this is a $$\partial$$-connected sum) and consider the solid torus alone. Here, if you push $$c$$ inside the solid torus, the complement will be diffeomorphic to $$T \times [0,1]$$ for a torus $$T$$, hence any Dehn surgery on one component will give you a solid torus back.