Independence of satellites of torus knots in the smooth concordance group

Pinzón-Caicedo, Juanita

doi:10.2140/gt.2017.21.3191

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Independence of satellites of torus knots in the smooth concordance group

Juanita Pinzón-Caicedo

Geometry & Topology 21 (2017) 3191–3211

DOI: 10.2140/gt.2017.21.3191

Abstract

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

Keywords

concordance, Whitehead double, instanton, satellite, Chern–Simons

Mathematical Subject Classification 2010

Primary: 57M25

Secondary: 57N70, 58J28

References

Publication

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

Authors

Juanita Pinzón-Caicedo
Department of Mathematics University of Georgia Athens, GA 30605 United States
http://alpha.math.uga.edu/~juanita/

Volume 21, issue 6 (2017)