Polynomial splittings of metabelian von Neumann rho-invariants

TL;DR

This paper demonstrates that vanishing von Neumann rho-invariants for connected sums of knots imply the same for individual knots, leading to new examples of linearly independent knots in the concordance group.

Contribution

It establishes a splitting property for metabelian von Neumann rho-invariants and constructs new linearly independent knots in the concordance group.

Findings

Vanishing rho-invariants split under connected sum with coprime Alexander polynomials.

New infinite family of linearly independent knots in the concordance group.

Provides a method to analyze knot concordance using metabelian representations.

Abstract

We show that if the connected sum of two knots with coprime Alexander polynomials has vanishing von Neumann rho-invariants associated with certain metabelian representations then so do both knots. As an application, we give a new example of an infinite family of knots which are linearly independent in the knot concordance group.