The Role Of Link Concordance In Knot Concordance

TL;DR

This paper investigates how infection operations with n-component string links can produce knots that cannot be generated by infections with fewer components, revealing new complexities in knot concordance.

Contribution

It demonstrates that certain knots obtained via n-component string link infections cannot be realized through (n-1)-component infections, highlighting limitations in previous infection-based constructions.

Findings

Existence of knots from n-component infections not realizable by (n-1)-component infections

Infection operations can produce distinct concordance classes of knots

Results extend understanding of knot concordance and infection techniques

Abstract

Satellite constructions on a knot can be thought of as taking some strands of a knot and then tying in another knot. Using satellite constructions one can construct many distinct isotopy classes of knots. Pushing this further one can construct distinct concordance classes of knots which preserve some algebraic invariants. Infection is a generalization of satellite operations, which has been previously studies. An infection by a string link can be thought of as grabbing a knot at multiple locations and then tying in a link. Cochran, Friedl and Techner showed that any algebraically slice knot is the result of infecting a slice knot by a string link [5]. In this paper we use the infection construction to show that there exist knots which arise from infections by $n$-component string links that cannot be obtained by infecting along $(n-1)$- component string links.