Splittings of von Neumann rho-invariants of knots

TL;DR

This paper investigates conditions under which the vanishing of certain knot concordance invariants, specifically von Neumann rho-invariants, can be split under connected sum, extending previous results and applying to specific knot examples.

Contribution

It provides a new sufficient condition involving higher-order Blanchfield forms for the splitting of rho-invariants, expanding the understanding of knot concordance obstructions.

Findings

Splitting property of rho-invariants under connected sum established.

Knots with vanishing Casson-Gordon invariants are shown not to be concordant to genus one knots.

Extension of previous results on Alexander polynomial coprimality in knot concordance.

Abstract

We give a sufficient condition under which vanishing property of Cochran-Orr-Teichner knot concordance obstructions splits under connected sum. The condition is described in terms of self-annihilating submodules with respect to higher-order Blanchfield linking forms. This extends results of Levine and the authors on distinguishing knots with coprime Alexander polynomials up to concordance. As an application, we show that the knots constructed by Cochran, Orr and Teichner as the first examples of nonslice knots with vanishing Casson-Gordon invariants are not concordant to any knot of genus one.