$$\frac{dB}{dS} = -\tau N = \tau(-N)$$ $$\frac{dT}{dS} = \kappa N$$
where $\tau$ is the torsion and $\kappa$ is the curvature.
$\frac{dT}{dS}$, by convention, is defined to be in the direction of $N$, and hence $\kappa$ is always positive. This also means that $\kappa$ is the magnitude of $\frac{dT}{dS}$.
But $\frac{dB}{dS}$ is defined to be in the direction of $N$ or $-N$; hence, $\tau$ can be positive or negative. If $\frac{dB}{dS}$ is taken to be in the direction of $N$, then $\tau = -\Big|\frac{dB}{dS}\Big|$ (always negative). If $\frac{dB}{dS}$ is taken to be in the direction of $-N$, then $\tau = \Big|\frac{dB}{dS}\Big|$ (always positive).
Is there any mistake in my conceptual understanding? Also, why do we have two separation signing conventions for curvature and torsion?