1. The 27 possible intrinsic symmetry groups of two-component links
- Author
-
Jason Cantarella, Jason Parsley, James Cornish, and Matt Mastin
- Subjects
Physics and Astronomy (miscellaneous) ,Astrophysics::High Energy Astrophysical Phenomena ,General Mathematics ,Astrophysics::Cosmology and Extragalactic Astrophysics ,Symmetry group ,Prime (order theory) ,Combinatorics ,Mathematics::Group Theory ,Mathematics - Geometric Topology ,Conjugacy class ,Computer Science (miscellaneous) ,FOS: Mathematics ,Link (knot theory) ,Mathematics ,Group (mathematics) ,lcsh:Mathematics ,link symmetry ,Geometric Topology (math.GT) ,two-component links ,lcsh:QA1-939 ,Mathematics::Geometric Topology ,Whitten group ,Chemistry (miscellaneous) ,Product (mathematics) ,57M25 ,symmetry group of knot ,Component (group theory) ,Homomorphism - 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.
- Published
- 2012
- Full Text
- View/download PDF