The Role Of Link Concordance In Knot Concordance

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.