For all n>0 there exists a homomorphism from the smooth concordance group of knots in dimension 2n+1 to an algebraically defined group g^Q. This algebraic concordance group splits as a direct sum of groups indexed by polynomials. For n>1 the homomorphism is injective, and this leads to what is called a primary decomposition theorem. In the classical dimension, the kernel of this homomorphism includes the smooth concordance group of topologically slice knots, T, which has become an important focus of research about smooth knot concordance. Here we will show that primary decompositions of T of a strong type cannot exist.

In more detail, it is shown that there exists a topologically slice knot K for which there is a factorization of its Alexander polynomial, Δ_K(t) = f₁(t)f₂(t), where f₁ and f₂ are relatively prime and each is the Alexander polynomial of a topologically slice knot, but K is not smoothly concordant to any connected sum K₁ # K₂ for which Δ_Ki(t) = f_i(t)^n_i for any nonnegative integers n_i.

This work was supported by a grant from the National Science Foundation, NSF-DMS-1505586.