I'm studying the article "An alternative proof of Lickorish–Wallace theorem" ([doi link](https://doi.org/10.1016/j.topol.2013.06.009))
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](https://mathoverflow.net/questions/84440/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.



Dehn surgery along primitive knot in 3-dimensional handlebody
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.


