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Operadic actions on long knots and $2$–string links

Etienne Batelier and Julien Ducoulombier

Algebraic & Geometric Topology 23 (2023) 833–882
Abstract

We realize the space of 2–string links as a free algebra over a colored operad denoted by 𝒮𝒞 (for “Swiss-cheese for links”). This result extends works of Burke and Koytcheff about the quotient of by its center, and is compatible with Budney’s freeness theorem for long knots. From an algebraic point of view, our main result refines Blaire, Burke and Koytcheff’s theorem on the monoid of isotopy classes of string links. Topologically, it expresses the homotopy type of the isotopy class of a 2–string link in terms of the homotopy types of the classes of its prime factors.

Keywords
space of knots, space of links, little cubes operad, Swiss-cheese operad, embedding, homotopy theory
Mathematical Subject Classification
Primary: 57R40
Secondary: 55P48, 55U40, 57M99
References
Publication
Received: 2 March 2021
Revised: 22 October 2021
Accepted: 10 November 2021
Published: 9 May 2023
Authors
Etienne Batelier
Mathematisches Institut
Universität Münster
Münster
Germany
Julien Ducoulombier
Université Sorbonne Paris Nord
Villetaneuse
France

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