Closed curves of constant torsion. II

In this note we show that there exist closed regular ${C^3}$ space curves $\alpha$ with curvature $\kappa > 0$ and nonzero constant torsion $\tau$ whose total torsion $\smallint_\alpha {\tau \;ds}$ is arbitrarily small. In so doing, we give another proof of the existence of closed curves of nonzero constant torsion. This note shows that Conjecture 2 in [2] is incorrect since the preceding statement is equivalent to the statement that there exist closed curves of constant torsion $\tau = 1$ whose length is arbitrarily small.