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...

There are some notes of Hatcher about characteristic submanifolds you should read. — Charlie Frohman, Jul 06 '19 at 15:40
But along this line, I indeed found a picture here: http://katlas.math.toronto.edu/caldermf/3manifolds/3manifolds.pdf (Me and my friend were just unable to think of an example... The example here is rather simple though...) — ah--, Jul 06 '19 at 16:10

score 2 · Answer 1 · answered Jul 07 '19 at 07:21

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.

score 1 · Answer 2 · answered Jul 21 '19 at 16:01

1

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.

answered Jul 21 '19 at 16:01

Solveit

838

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

2 Answers2