Independence of satellites of torus knots in the smooth concordance group

Pinzón-Caicedo, Juanita¹

¹ Department of Mathematics, University of Georgia, Athens, GA 30605, United States

Geometry & topology, Tome 21 (2017) no. 6, pp. 3191-3211.

Résumé

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

Détail

DOI : 10.2140/gt.2017.21.3191

Classification : 57M25, 57N70, 58J28
Keywords: concordance, Whitehead double, instanton, satellite, Chern–Simons

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