Is there a way to visualize an incompressible torus in a 3-manifold?

Question

I would really like to understand how an incompressible torus looks like, but could not think of a picture of it for a long time...

Answer 1

Take a solid torus $\hat{T}$ in $S^3$, let $M=S^3- \hat{T}$. Then, unless $\hat{T}$ is unknotted in $S^3$, the boundary of $M$ will be an incompressible torus in $M$. If you want to get an incompressible torus in a closed manifold, glue two such manifolds $M_1, M_2$ along their boundary tori.

Answer 2

Consider the 3-torus as a cube with opposing faces identified. A cross-section of the 3-torus taken in the most obvious way is an incompressible 2-torus.