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

TL;DR

This paper introduces a new class of Von Neumann rho-invariants for knots, providing tools to detect infinite order and torsion elements in the knot concordance group, with explicit computations for certain twist knots.

Contribution

It defines a family of rho-invariants that are homomorphisms on a subgroup of the concordance group and develops a computable bound to identify infinite order and torsion in knots.

Findings

Computed bounds for the invariants for specific twist knots.

Identified an infinite set of linearly independent knots of algebraic order 2.

Demonstrated the invariants' effectiveness in detecting infinite order in the 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 the real line. 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 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 .