Topology Vol. 24. No. I. pp. 15-23. 1985. 0040-9383/85 S3.00+ .oO Pnnted m Gmt Bntain. Pcrgamon Press Ltd. COUNTING TRITANGENT PLANES OF SPACE CURVES THOMAS BANCHOFF. TERENCE GAFFNEY and CLINT MCCRORY* (Receioed 20 July 1983) LET C be a smooth simple closed curve in Iw3. A tritangent plane of C is a plane in W3 which is tangent to C at exactly three points. A stall x of C is a point of C at which the torsion of C is zero. We will say that a stall x is transoerse if the curvature of C is non-zero at x, the derivative of the torsion of C is non-zero at x, and the osculating plane P of C at x is transverse to C away from x. If x is a transverse stall of C then an interval of C about x lies on one side of the osculating plane P of C at x, so P intersects Cat an even number 2n of points other than x. The integer n = n(x, C) is the index of the transverse stall x of C. Let Coc(S’, rW3) be the space of C” maps ~1: S’ -+ W3with the Whitney topology. THEOREM. There is an open dense subset A ofCm (S I, rW3) such that ifa E A then C = a(S ‘) is a simple curve with a$nite number T(C) of tritangent planes and afinite number of stalls Xl,. . . . ) xk, all of which are transverse. 1 a E A then T(C) E i n(x,, C) (mod 2). i=l An explicit description of the set A is given in Section 1 below. This theorem generalizes the result of M. Freedman that a generic smooth closed space curve with nonvanishing torsion has an even number of tritangent planes [43. To prove the theorem, we consider the classical dual surface C* consisting of all planes in Iw3 tangent to C. We use the theory of singularities of maps to analyze the singularities of C*. The tritangent planes of C correspond to triple points of C*, and the stalls of C correspond to swallowtail points of C* (cf. [3]). Then we count the triple points of C* using a generalization of the techniques of [l]. As this paper was in its final stages of preparation, we received a letter from Tetsuya Ozawa announcing an integer formula for the triple tangent planes. Properly indexed, the sum of the tritangent planes equals the sum of the stalls, each stall counting + n(x, C) times. Ozawa also has a formula relating the stalls and osculating planes which are tangent to the curve at two points. 81. Let A be the subset of Cm(S1, following seven conditions. Local conditions: THE GENERIC SET A. H3) consisting of all maps a: S’ -, lR3 satisfying the 1. a is regular, i.e. a’(t) # 0 for all t ES’. 2. The curvature K of a is never zero. 3. The zeros of the torsion r of a are nondegenerate, i.e., T(t) = 0 implies r’(t) # 0. Global conditions: 4. a is injective. 5. No plane P in Iw3 is tangent to a at more than three points. If P is tangent to a at three points then the three points are non-collinear. l Supported in part by NSF grants MC579-01310. MC579-04905. MCS-8102759. 15