Von Neumann rho invariants and torsion in the topological knot concordance group

Abstract

We discuss an infinite class of metabelian Von Neumann $\rho$–invariants. Each one is a homomorphism from the monoid of knots to $ℝ$. In general they are not well defined on the concordance group. Nonetheless, we show that they pass to well defined homomorphisms from the subgroup of the concordance group generated by anisotropic knots. Thus, the computation of even one of these invariants can be used to conclude that a knot is of infinite order. We introduce a method to give a computable bound on these $\rho$–invariants. Finally we compute this bound to get a new and explicit infinite set of twist knots which is linearly independent in the concordance group and whose every member is of algebraic order 2.