What is the fundamental group of the $3$-manifold bounded by a genus-$2$ torus?

Question

As in the question, let $X$ be the $3$-manifold bounded by $\partial X$ which is a torus of genus $2$.What is $\pi_1(X)$?

I noted that $\pi_1(\mathbb{R} ^3\backslash X) \cong \langle a, b | aba^{-1} b^{-1} \rangle$ but I cannot see any commutator relation in $\pi_1(X) $. However I also know that $\pi_1(\partial X) $ has a relation between its four generators. I just cannot see why that does not translate to $\pi_1(X)$.

Answer 1

Your manifold deformation retracts to a figure-eight (or, perhaps more easily, to a figure-$\theta$). Meaning it has fundamental group $\langle a,b\rangle$.