Geometry & Topology

Independence of satellites of torus knots in the smooth concordance group

Juanita Pinzón-Caicedo

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Abstract

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.

Article information

Source
Geom. Topol., Volume 21, Number 6 (2017), 3191-3211.

Dates
Received: 16 January 2015
Revised: 13 October 2016
Accepted: 25 December 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510859319

Digital Object Identifier
doi:10.2140/gt.2017.21.3191

Mathematical Reviews number (MathSciNet)
MR3692965

Zentralblatt MATH identifier
1373.57024

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57N70: Cobordism and concordance 58J28: Eta-invariants, Chern-Simons invariants

Keywords
concordance Whitehead double instanton satellite Chern–Simons

Citation

Pinzón-Caicedo, Juanita. Independence of satellites of torus knots in the smooth concordance group. Geom. Topol. 21 (2017), no. 6, 3191--3211. doi:10.2140/gt.2017.21.3191. https://projecteuclid.org/euclid.gt/1510859319


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