[Paper Review] VON NEUMANN RHO INVARIANTS AS OBSTRUCTIONS TO TORSION IN THE TOPOLOGICAL KNOT CONCORDANCE GROUP

Christopher William Davis|arXiv (Cornell University)|Oct 24, 2010

Geometric and Algebraic Topology19 references3 citations

TL;DR

This paper introduces an infinite family of metabelian Von Neumann ρ-invariants that serve as obstructions to torsion in the topological knot concordance group. By showing these invariants are well-defined on the subgroup generated by anisotropic knots, the authors prove that computing even one such invariant can demonstrate a knot has infinite order, and they explicitly construct a linearly independent infinite set of twist knots, each of algebraic order 2, using a computable bound on the invariants.

ABSTRACT

Abstract. We discuss an infinite class of metabelian Von Neumann ρ-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 ρ-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. 1.

Motivation & Objective

• To investigate the behavior of metabelian Von Neumann ρ-invariants in the context of knot concordance, particularly their applicability to detecting non-torsion elements.
• To determine under what conditions these ρ-invariants are well-defined on the concordance group, especially in relation to anisotropic knots.
• To develop a computable method for bounding these ρ-invariants to enable explicit calculations.
• To apply the bounds to construct an explicit, infinite family of twist knots that are linearly independent in the concordance group and of algebraic order 2.

Proposed method

• The authors define an infinite class of metabelian Von Neumann ρ-invariants as homomorphisms from the monoid of knots to the real numbers.
• They establish that these invariants descend to well-defined homomorphisms on the subgroup of the concordance group generated by anisotropic knots.
• A novel method is introduced to compute an effective upper bound on the ρ-invariants using algebraic and analytic techniques from von Neumann algebra and knot theory.
• The bound is applied to a family of twist knots, enabling explicit computation of the invariants and verification of their linear independence.

Experimental results

Research questions

• RQ1 Under what conditions are metabelian Von Neumann ρ-invariants well-defined on the concordance group?
• RQ2 Can these ρ-invariants be used to detect knots of infinite order in the topological concordance group?
• RQ3 What is a computable upper bound for these ρ-invariants that allows explicit calculations?
• RQ4 Can the bound be used to construct an infinite linearly independent set of twist knots of algebraic order 2?

Key findings

• The ρ-invariants are well-defined on the subgroup of the concordance group generated by anisotropic knots, enabling their use as obstructions to torsion.
• Computing a single ρ-invariant can prove that a knot has infinite order in the topological concordance group.
• A computable upper bound is derived for the ρ-invariants, making explicit calculations feasible.
• An explicit infinite set of twist knots is constructed that are linearly independent in the concordance group.
• Every member of this infinite set is shown to have algebraic order 2.
• The construction demonstrates that the ρ-invariants detect non-torsion elements in the concordance group through concrete, computable examples.