Intrinsic symmetry groups of links

TL;DR

This paper investigates the intrinsic symmetry groups of links in the 3-sphere, providing counterexamples that show not all subgroups of the symmetric group can be realized as such for links with more than five components.

Contribution

It demonstrates that certain subgroups, like the alternating group, cannot be realized as intrinsic symmetry groups of links when the number of components exceeds five.

Findings

Counterexamples for n > 5 components showing not all subgroups are realizable

The alternating group cannot be an intrinsic symmetry group for links with more than five components

Advances understanding of the limitations of link symmetry groups

Abstract

The set of isotopy classes of ordered n-component links in the 3-sphere is acted on by the symmetric group via permutation of the components. The intrinsic symmetry group of the link, S(L), 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 the symmetric group arises as an intrinisic symmetry group of some link has remained open. We provide counterexamples; in particular, if n > 5, then there does not exist an n-component link L for which S(L) is the alternating group.