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The 27 Possible Intrinsic Symmetry Groups of Two-Component Links

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Author(s): Jason Cantarella | James Cornish | Matt Mastin | Jason Parsley

Journal: Symmetry
ISSN 2073-8994

Volume: 4;
Issue: 1;
Start page: 129;
Date: 2012;
Original page

Keywords: two-component links | symmetry group of knot | link symmetry | Whitten group

ABSTRACT
We consider the “intrinsic” symmetry group of a two-component link L, defined to be the image ∑(L) of the natural homomorphism from the standard symmetry group MCG(S3, L) to the product MCG(S3) × MCG(L). This group, first defined by Whitten in 1969, records directly whether L is isotopic to a link L′ obtained from L by permuting components or reversing orientations; it is a subgroup of Γ2, the group of all such operations. For two-component links, we catalog the 27 possible intrinsic symmetry groups, which represent the subgroups of Γ2 up to conjugacy. We are able to provide prime, nonsplit examples for 21 of these groups; some are classically known, some are new. We catalog the frequency at which each group appears among all 77,036 of the hyperbolic two-component links of 14 or fewer crossings in Thistlethwaite’s table. We also provide some new information about symmetry groups of the 293 non-hyperbolic two-component links of 14 or fewer crossings in the table.
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