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

Davis, Christopher William

doi:10.2140/agt.2012.12.753

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

Christopher William Davis

Algebraic & Geometric Topology 12 (2012) 753–789

DOI: 10.2140/agt.2012.12.753

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.

Keywords

knot concordance, rho-invariants

Mathematical Subject Classification 2010

Primary: 57M25, 57M27, 57N70

References

Publication

Received: 16 February 2011

Revised: 28 September 2011

Accepted: 28 September 2011

Published: 12 April 2012

Authors

Christopher William Davis
Department of Mathematics Rice University 6100 Main St Houston TX 77005 USA
http://math.rice.edu/~cwd1/

Volume 12, issue 2 (2012)