Let $T=D^2 \times S^1$ be a solid torus, where $D^2$ is a 2-dimensional disk and $S^1$ is a circle.

Suppose we have another solid torus $T'$ and we have a homeomorphism $f$ sending a meridian $\partial D^2\times \{*\} \subset T$ to a longitude $\{*\}\times S^1\subset T'$.

Then gluing $T, T'$ along the boundary torus via the homeomorphism $f$ results in a 3-sphere.

Instead of thinking another torus $T'$, we can think about the complement of $T'$ in $S^3$, which is also a solid torus.

(Now I am getting confused.)

Then, the above homeomorphism corresponds to a homeomorphism sending a meridian to a meridian, or orientation reversed meridian?

So my questions are:

- Once we choose a meridian and longitude of a torus, is there a canonical choice for a meridian and longitude of the compliment torus?
Let $(a, b)\in \mathbb{Z} \times \mathbb{Z}$ represent a curve in a torus winding $a$ times in the meridian direction and $b$ times in the longitude direction.

I read somewhere that if we turn a torus inside out, the homeomorphism sending a meridian to $(m,l)$ corresponds to the homeomorphism sending a meridian to $(-l, -m)$.Is this right? Or, for this to be true, what choice should I make? (orientation, meridian, longitude.)

I appreciate any help.