#### Volume 21, issue 6 (2017)

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Independence of satellites of torus knots in the smooth concordance group

### Juanita Pinzón-Caicedo

Geometry & Topology 21 (2017) 3191–3211
##### Abstract

The main goal of this article is to obtain a condition under which an infinite collection $\mathsc{ℱ}$ of satellite knots (with companion a positive torus knot and pattern similar to the Whitehead link) freely generates a subgroup of infinite rank in the smooth concordance group. This goal is attained by examining both the instanton moduli space over a $4$–manifold with tubular ends and the corresponding Chern–Simons invariant of the adequate $3$–dimensional portion of the $4$–manifold. More specifically, the result is derived from Furuta’s criterion for the independence of Seifert fibred homology spheres in the homology cobordism group of oriented homology $3$–spheres. Indeed, we first associate to $\mathsc{ℱ}$ the corresponding collection of $2$–fold covers of the $3$–sphere branched over the elements of $\mathsc{ℱ}$ and then introduce definite cobordisms from the aforementioned covers of the satellites to a number of Seifert fibered homology spheres. This allows us to apply Furuta’s criterion and thus obtain a condition that guarantees the independence of the family $\mathsc{ℱ}$ in the smooth concordance group.

##### Keywords
concordance, Whitehead double, instanton, satellite, Chern–Simons
##### Mathematical Subject Classification 2010
Primary: 57M25
Secondary: 57N70, 58J28