Obstructing Four-Torsion in the Classical Knot Concordance Group

In his classification of the knot concordance groups, Levine [L1] defined the algebraic concordance groups, G±, of Witt classes of Seifert matrices and a homomorphism from the odd-dimensional knot concordance groups C4n±1 to G±. The homomorphism is induced by the function that assigns to a knot an associated Seifert matrix: it is an isomorphism on Ck, k ≥ 5; on C3 it is injective, onto an index 2 subgroup in smooth category and surjective in the topological locally flat setting; for k = 1 it is surjective. However, Casson and Gordon [CG1,CG2] proved that on C1 the kernel is nontrivial. (Casson and Gordon’s original work applied in the smooth setting, but their results are now known to hold in the topological locally flat setting as well, a fact that follows from the existence of normal bundles in topological 4-manifolds, [FQ]. Similarly, our work applies in both categories.) Later, Jiang [J] extended Casson and Gordon’s work to prove that the kernel of Levine’s homomorphism is infinitely generated. Levine [L2] also proved that G± is isomorphic to an infinite direct sum, G± ∼= Z ∞ ⊕ Z∞2 ⊕ Z ∞ 4 . There is 2-torsion in C1 arising from amphicheiral knots, but beyond this little is known concerning torsion in C1. Fox and Milnor [FM], in the paper in which knot concordance is defined, made this observation concerning amphicheiral knots and asked if there is torsion of any order other than 2. This question reappears as problem 1.32 of [K1, K2]. In a different direction, in 1977 Gordon [G] (see also [K2], problem 1.94) asked whether every order 2 class in C1 is represented by an amphicheiral knot; as of yet the only result bearing on this question is the observation that in higher dimensions the answer is no, and in dimension 3 there are order 2 classes in G− that cannot be represented by order 2 knots in C1 [CM]. In this paper we will use Casson-Gordon invariants to derive results concerning 4-torsion in the classical knot concordance group. That Casson-Gordon invariants can be used to show that an individual knot that is of algebraic order 4 is not of order 4 in concordance is not surprising, though the examples presented here are the first; that the method applies to the knot 77 is pleasing in that this is the first knot identified by Levine [L2] as a candidate to be of order 4. It is surprising that the obstructions we find depend only on