Independence of Satellites of Torus Knots in the Smooth Concordance Group

Abstract

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.

Cited by in corpus (4)

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• Chern-Simons Functional and the Homology Cobordism Group
• An adjunction inequality for the Bauer-Furuta type invariants, with applications to sliceness and 4-manifold topology
• 3-manifolds without any embedding in symplectic 4-manifolds