The 27 possible intrinsic symmetry groups of two-component links

Abstract

We consider the "intrinsic" symmetry group of a two-component link L, defined to be the image Σ(L) of the natural homomorphism from the standard symmetry group MCG(S 3, L) to the product MCG(S 3) × 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 Γ 2, the group of all such operations. For two-component links, we catalog the 27 possible intrinsic symmetry groups, which represent the subgroups of Γ 2 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. © 2012 by the authors.