Curves with no tritangent planes in space and their convex envelopes

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M. H. Freedman ([3]) proved that for a generic subset of closed curves in ~ 3 with nonvanishing curvature and torsion the number of t r i tangent planes is even and finite. He also guessed, for each even number s _> 0, the existence of an open subset A8 of closed curves with nonvanishing curvature and torsion such tha t each curve in A8 has exact ly s t r i t angent planes. A question tha t can be asked in this context is: Which curves with nonvanishing curvature and torsion have no t r i tangent planes? An example of such a curve is given by the (1,2)-curve on the torus with rat io a, 3 < a < 5 (see [2]). For a generi c curve, we give a pa r t i a l answer to this question here by finding a necessary condit ion for the curve to have no t r i tangent planes. In fact, we find tha t a sufficient condi t ion for a generic curve to have at least two tr i tangencies (par t icu lar case of which is a t r i tangent plane) is re la ted to the s t ructure of its convex envelope; we also extend this result to a wider class of curves by using l imit ing arguments. Elsewhere [6], we apply similar methods to a discussion of lower bounds for the number of torsion zero points (or vertices) of a curve in ~ 3 .