Primary decomposition of knot concordance and von Neumann rho-invariants

TL;DR

This paper investigates the primary decomposition of the knot concordance group using solvable filtrations and von Neumann rho-invariants, providing evidence for conjectures about nonslice knots with coprime Alexander polynomials.

Contribution

It establishes conditions under which knots with coprime Alexander polynomials have vanishing rho-invariants, advancing understanding of knot concordance and primary decomposition.

Findings

Connected sum of n-solvable knots with coprime Alexander polynomials being slice implies each has vanishing rho-invariants.

Nonslice knots with coprime Alexander polynomials are likely not concordant.

Certain nonslice knots with vanishing Casson-Gordon invariants are not concordant to knots with coprime Alexander polynomials.

Abstract

We address the primary decomposition of the knot concordance group in terms of the solvable filtration and higher-order von Neumann $\rho$-invariants by Cochran, Orr, and Teichner. We show that for a nonnegative integer n, if the connected sum of two n-solvable knots with coprime Alexander polynomials is slice, then each of the knots has vanishing von Neumann $\rho$-invariants of order 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 is one of Cochran-Orr-Teichner's knots which are the first examples of nonslice knots with vanishing Casson-Gordon invariants, then K is not concordant to any knot with Alexander polynomial coprime to that of K.