KNOTS WHICH ARE NOT CONCORDANT TO THEIR REVERSES By CHARLES LIVINGSTON

IF K is an oriented knot in S, the reverse of K, K*, is the knot K with its orientation reversed. (This has traditionally been called the inverse of K. We call it the reverse to distinguish it from the inverse to K in the knot concordance group, denoted by -K and represented by the mirror image of K with orientation reversed.) Fox [3] asked for an example of a knot which is not isotopic to its reverse. Trotter provided the first examples in [8]. In this paper we will give examples of knots which are not concordant to their reverses. Finding such an example is difficult on two accounts. A knot and its reverse have equivalent images in the algebraic concordance group defined by Levine [6]. This follows from the fact that if V is a Seifert matrix for K, then V is a Seifert matrix for K*. In [7] Levine defined a complete set of invariants for the algebraic concordance group, none of which are changed by taking the transpose of an element. A second difficulty is that the branched covers of S branched along K and K* are identical. Hence a direct application of the techniques of Casson and Gordon [1, 2] will not work. Our approach is to use the refinement of the Casson-Gordon technique which was developed by Gilmer [4]. Throughout this work we will use the results and notation of [4]. Thanks are due to Pat Gilmer for many informative and helpful conversations, and to Cameron Gordon for pointing out this problem.