Splitting the Concordance Group of Algebraically Slice Knots

As a corollary of work of Ozsváth and Szabó, it is shown that the classical concordance group of algebraically slice knots has an infinite cyclic summand and in particular is not a divisible group. Let A denote the concordance group of algebraically slice knots, the kernel of Levine’s homomorphism φ : C → G, where C is the classical knot concordance group and G is Levine’s algebraic concordance group [6]. Little is known about the algebraic structure of A: it is countable and abelian, Casson and Gordon [2] proved that A is nontrivial, Jiang [5] showed it contains a subgroup isomorphic to Z∞, and the author [7] proved that it contains a subgroup isomorphic to Z∞ 2 . We add the following theorem, a quick corollary of recent work of Ozsváth and Szabó [8]. Theorem 1. The group A contains a summand isomorphic to Z and in particular A is not divisible. Proof. In [8] a homomorphism τ : C → Z is constructed. We prove that τ is nontrivial on A. The theorem follows since, because Im(τ) is free, there is the induced splitting, A ∼= Im(τ) ⊕ Ker(τ). No element representing a generator of Im(τ) is divisible. According to [8], |τ(K)| ≤ g4(K), where g4 is the 4–ball genus of a knot, and there is the example of the (4, 5)–torus knot T for which τ(T ) = 6. We will show that there is a knot T ∗ algebraically concordant to T with g4(T ∗) < 6. Hence, T#− T ∗ is an algebraically slice knot with nontrivial τ , as desired. Recall that T is a fibered knot with fiber F of genus (4 − 1)(5 − 1)/2 = 6. Let V be the 12 × 12 Seifert matrix for T with respect to some basis for H1(F ). The quadratic form q(x) = xV x on Z is equal to the form given by (V + V )/2. Using [3] the signature of this symmetric bilinear form can be computed to be 8, so q is indefinite, and thus by Meyer’s theorem [4] there is a nontrivial primitive element z with q(z) = 0. Since z is primitive, it is a member of a symplectic basis for H1(F ). Let V ∗ be the Seifert matrix for T with respect to that basis. The canonical construction of a Seifert surface with Seifert matrix V ∗ ([9], or see [1]) yields a surface such that z is represented by a simple closed curve on F ∗ that is unknotted in S. Hence, F ∗ can be surgered in the 4–ball to show that g4(T ∗) < 6. Since T ∗ and T have the same Seifert form, they are algebraically concordant. Date: March 22, 2008.