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Intrinsic symmetry groups of links

Charles Livingston

Algebraic & Geometric Topology 23 (2023) 2347–2368
Abstract

The set of isotopy classes of ordered n–component links in S3 is acted on by the symmetric group 𝕊n via permutation of the components. The subgroup 𝕊(L) 𝕊n is defined to be the set of elements in the symmetric group that preserve the ordered isotopy type of L as an unoriented link. The study of these groups was initiated in 1969, but the question of whether or not every subgroup of 𝕊n arises as such an intrinsic symmetry group of some link has remained open. We provide counterexamples; in particular, if n 6, then there does not exist an n–component link L for which 𝕊(L) is the alternating group 𝔸n.

Keywords
link theory, symmetry group of a link
Mathematical Subject Classification
Primary: 57K10
References
Publication
Received: 1 November 2021
Revised: 13 January 2022
Accepted: 3 February 2022
Published: 25 July 2023
Authors
Charles Livingston
Department of Mathematics
Indiana University
Bloomington, IN
United States

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