2-bridge knots are small
Here's a cool fact due to Hatcher and W. Thurston ( article here ): 2-bridge knots are small . I never realized the proof was so simple (and clever), so I wanted to write about it. I'll briefly explain what small means and then sketch the proof (which is actually fairly short). Remark: They prove something much more general. They are characterizing incompressible surfaces in the complement of 2-bridge knots, but I won't discuss that here. Background: An embedded surface in a three-manifold is incompressible if the inclusion map induces an injection on $\pi_1$. For example, the boundary of a handlebody is not incompressible but $S^1 \times \Sigma_g$ has many incompressible surfaces: $pt \times \Sigma_g$ or $S^1 \times \gamma$ for $\gamma$ non-trivial in $\pi_1(\Sigma_g)$. A neat fact is that a knot in $S^3$ is non-trivial if and only if the boundary of the exterior is incompressible. A knot is small if there are no closed incompressible surfaces in the ext