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Symmetry 2012, 4(1), 129-142; doi:10.3390/sym4010129

The 27 Possible Intrinsic Symmetry Groups of Two-Component Links

1
University of Georgia, Mathematics Department, Boyd GSRC, Athens, GA 30602, USA
2
Wake Forest University, Mathematics Department, Box 7388, Winston-Salem, NC 27109, USA
*
Author to whom correspondence should be addressed.
Received: 13 January 2012 / Revised: 7 February 2012 / Accepted: 9 February 2012 / Published: 17 February 2012
(This article belongs to the Special Issue Symmetry and Beauty of Knots)
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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.
Keywords: two-component links; symmetry group of knot; link symmetry; Whitten group two-component links; symmetry group of knot; link symmetry; Whitten group
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This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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MDPI and ACS Style

Cantarella, J.; Cornish, J.; Mastin, M.; Parsley, J. The 27 Possible Intrinsic Symmetry Groups of Two-Component Links. Symmetry 2012, 4, 129-142.

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