I was thinking the fact that if two genus $1$ handlebodies (solid tori) are glued via an orientation preserving homeomorphism of boundaries, the resulting manifold depends only on (up to isotopy) the image of a neighborhood of a meridian.
Let $\tau$ is a Dehn twist along a meridian. According to the above fact, the resulting manifold via $\tau$ is same as the manifold obtained via the identity homeomorphism of the boundaries since they act identity on a neighborhood of a meridian.
I'm convinced by this argument but my brain (intuition/ geometric intuition?) doesn't accept the fact. Even we do a twist, they don't change the result? So I did several experiments. One of them is as follows.
Suppose we put an annulus in a handlebody of genus $1$ along a longitude. (Homology of the annulus is not zero.) Then by the identity homeomorphism, we have $S^1 \times S^2$ and the annulus is in it as it is.
If we glue handledodies by $\tau$, we should have $S^1 \times S^2$ by the fact. But it seems for me, the annulus is "twisted" along a meridian.
Is there an isotopy between these two such that a twisted annulus is mapped to non-twisted annulus?
Or If we consider a subset in handlebody, the above fact doesn't hold true anymore?
Thank you in advance.