Primary decomposition of knot concordance and von Neumann rho-invariants

Abstract

We address the primary decomposition of the knot concordance group in terms of the solvable filtration and higher order von Neumann ρ \rho -invariants by Cochran, Orr, and Teichner. We show that for a non-negative integer n n , if the connected sum of two n n -solvable knots with coprime Alexander polynomials is slice, then each of the knots has vanishing von Neumann ρ \rho -invariants of order n n . This gives positive evidence for the conjecture that nonslice knots with coprime Alexander polynomials are not concordant. As an application, we show that if K K is one of Cochran–Orr–Teichner’s knots which are the first examples of nonslice knots with vanishing Casson–Gordon invariants, then K K is not concordant to any knot with Alexander polynomial coprime to that of K K .