Counting normals to closed curves in $\mathbb{R}^3$

We prove the following results: (1) For every generic closed smooth curve in $\mathbb{R}^3$, {whose torsion has a constant sign}, there is a point with at least $6$ emanating normals to the curve. If the curve is knotted, and torsion has a constant sign, there is a point with at least $8$ emanat