Independence of Satellites of Torus Knots in the Smooth Concordance Group

TL;DR

This paper establishes a condition under which an infinite family of satellite knots, with positive torus knot companions and specific patterns, are linearly independent in the smooth concordance group by using gauge theory and cobordism techniques.

Contribution

It introduces a new criterion for the independence of satellite knots in the smooth concordance group based on instanton moduli spaces and Furuta's cobordism criterion.

Findings

Infinite satellite knots can generate a subgroup of infinite rank in the concordance group.

The criterion applies to satellites with companions as positive torus knots and patterns similar to the Whitehead link.

The method involves analyzing 2-fold branched covers and Seifert fibered homology spheres.

Abstract

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