Effect of adding a constant to the torsion of a 3D curve

Question

Let $\gamma$ be an arc-length parametrized curve in $\mathbb{R}^3$. Let say I add a constant to the torsion of $\gamma$ and let $\widetilde{\gamma}$ be the curve associated to the curvature of $\gamma$ and this new torsion. Is there an easy way to characterize $\widetilde{\gamma}$ in terms of $\gamma$? In other words, is there a map that links $\gamma$ to $\widetilde{\gamma}$?