Closed curves of constant torsion. II
Abstract: In this note we show that there exist closed regular space curves with curvature and nonzero constant torsion whose total torsion 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  is incorrect since the preceding statement is equivalent to the statement that there exist closed curves of constant torsion whose length is arbitrarily small.
-  Werner Fenchel, On the differential geometry of closed space curves, Bull. Amer. Math. Soc. 57 (1951), 44–54. MR 0040040, https://doi.org/10.1090/S0002-9904-1951-09440-9
-  Joel L. Weiner, Closed curves of constant torsion, Arch. Math. (Basel) 25 (1974), 313–317. MR 0346712, https://doi.org/10.1007/BF01238680
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Keywords: Closed space curves, constant torsion, total torsion, binormal indicatrix, Peano direction
Article copyright: © Copyright 1977 American Mathematical Society