The fundamental group of any two-bridge knot K in $\mathbb{S}^3$ has a presentation with two generators and one relation.

On the other hand, it's possible to provide a CW-complex with only one 0-cell and no 3-cell on which the knot complement deformation retracts.

Is it reasonable to hope that we should provide such a CW-complex with only two 1-cell and one 2-cell, with boundary given by the relator of the $\pi_1$, on which $\mathbb{S}^3 \setminus V(K)$ retracts (notice it's possible for torus knots)? Or at least homotopically equivalent?

Thanks, any help would be appreciate.