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1985 (Publication #46)

Counting Tritangent Planes of Space Curves (with T. Gaffney and C. McCrory)

Topology 34 (1985), 15-24.

View a review of this article at MathSciNet


If a plane is tangent to a smooth simple closed space curve, C, the plane is said to be a tritangent plane. A stall of C is a point of zero torsion, and a stall is said to be transverse if the curvature is non-zero, the derivative of the torsion is non-zero, and the osculating plane is transverse to C away from the stall. So if x is a transverse stall of C, then an interval of C lies on one side of the osculating plane, and so the plane intersects C at an even number of points (other than x), say 2n. The number n is said to be the index of x.

This paper presents a formula for the number of tritangent planes of a curve in terms of the index the stalls of the curve.