This question is somewhat a continuation of the question Gluing a solid torus to a solid torus with annulus inside.
If we consider a genus one handlebody $U$ with an (nontwisted)annulus inside the handlebody so that its first homology is not zero in the handlebody.
Let $V$ be another genius one handlebody. Let $f$ be a Dehn twist about a meridian of the boundary of $U$. We think this map $f$ as a homeomorphism from the boundary of $U$ to the boundary of $V$.
Let $U \cup_f V$ be a result of gluing via $f$, which is isotopic to $S^1\times S^2$. Then the annulus is in $U \cup_f V$ somehow.
We obtain the same manifold as follows. First, we do a twist on $U$ along a meridian (extending the Dehn twist) and then glue it with $V$ via an identity map of the boundaries.
Then in this case if we look at the annulus in the resulting manifold, it look like locally (in the $U$) a twisted annulus since we did twist first.
But in the first case, the annulus is not twisted locally.
So my questions are;
- Is there a canonical way to define a twist or framing in a 3-manifold?
- What is a good way to look at knots or annulus in a 3-manifold? (as the above example, we need some method to look at knots in a 3-manifold.)
Any help or references are appreciated. Thank you in advance.