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Independence of satellites of torus knots in the smooth concordance group

Juanita Pinzón-Caicedo

Geometry & Topology 21 (2017) 3191–3211

The main goal of this article is to obtain a condition under which an infinite collection 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 the corresponding collection of 2–fold covers of the 3–sphere branched over the elements of 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 in the smooth concordance group.

concordance, Whitehead double, instanton, satellite, Chern–Simons
Mathematical Subject Classification 2010
Primary: 57M25
Secondary: 57N70, 58J28
Received: 16 January 2015
Revised: 13 October 2016
Accepted: 25 December 2016
Published: 31 August 2017
Proposed: Peter Teichner
Seconded: András I. Stipsicz, Rob Kirby
Juanita Pinzón-Caicedo
Department of Mathematics
University of Georgia
Athens, GA 30605
United States