TITLE: When does one obtain different 3-manifolds by pasting two tori? 
QUESTION [6 upvotes]: Consider a compact solid torus $T$ and a diffeomorphic copy of it $T' \subset T$ embedded in the interior of $T$ in such a way that it makes two turns around the central circle of $T$.

I would like to consider the compact boundaryless 3-manifolds $M_f$ obtained from the difference $M = T \setminus \mathring{T'}$ of $T$ and the interior of $T'$ by gluing the two torus boundaries $T_1$ and $T_2$ of $M$ with a diffeomorphism $f$.
If two diffeomorphisms $f,f'$ are isotopic then $M_f$ and $M_{f'}$ are diffeomorphic.   
Question:  Is this the only way one can have $M_f$ diffeomorphic to $M_{f'}$?  If not, is there a characterization of when one obtains diffeomorphic manifolds?  Is there a way to know what Thurston geometry the manifold $M_f$ will have?
I've spent some time trying to calculate the fundamental group of the resulting manifolds via Van Kampen's theorem but I didn't get very far.

REPLY [5 votes]: Here's a way to get different gluing maps giving the same glued-up manifold.
There's an annulus embedded in M that meets each boundary in a closed curve (on the outer torus $T_1$ this closed curve runs twice around the longitudinal direction).  There's a self-diffeomorphism of $M$ supported near the annulus that restricts to a Dehn twist $D_1$ on the closed curve in $T_1$ and to a Dehn twist $D_2$ on the closed curve on $T_2$.  It follows that if $f : T_1 \rightarrow T_2$ is a gluing map, then for $f' = D_2 \circ f \circ D_1^{-1}$ we have $M_f = M_{f'}$.