Conditions that torsion is zero in a space curve What are the conditions for torsion to be zero other than having a plane curve?
The only thing I can thing of is an equation that have the torsion that cancels out each other.

As has been said, the curve is planar iff the torsion is zero. This is clear if you look at the Frenet formulas. For fun, you can get from those a formula that the curve $\alpha: \mathbb{R} \rightarrow \mathbb{R}^3$ must satisfy. We need
$$
\frac{d\mathbf{N}}{ds} + \kappa\mathbf{T} = 0.
$$
Assuming $\alpha$ is parameterized by arc length, this could be written
$$
\frac{d}{ds}\left(\frac{1}{\kappa}\frac{d^2\alpha}{ds^2}\right) + \kappa\frac{d\alpha}{ds} = 0
$$
