Abstract
We present an elementary derivation of the “intrinsic” symmetry groups for links of 8 or fewer crossings. We show that standard invariants are enough to rule out all potential symmetries outside the symmetry group of the group of the link for all but one of these links and present explicit isotopies generating the symmetry group for every link.
Highlights
The symmetry group of a link L is defined to be the mapping class group MCG(L) (or Sym(L)) of the pair (S3, L). The study of this symmetry group is a classical topic in knot theory, and these groups have been computed for prime knots and links in several ways
The symmetry group Σ(L) of a given link must form a subgroup of Γ2
Question 7.1 Do all 27 nonconjugate subgroups of Γ2 appear as the symmetry group of some link? Of some prime, non-split link?
Summary
The symmetry group of a link L is defined to be the mapping class group MCG(L) (or Sym(L)) of the pair (S3, L). Weeks and Henry used the program SnapPea to compute the symmetry groups for hyperbolic knots and links of 9 and fewer crossings [3]. It is worth noting that SnapPea is a large and complicated computer program, and while its results are accurate for the links in our table, it is always worthwhile to have alternate proofs for results that depend essentially on nontrivial computer calculations In this spirit, the present paper presents an elementary and explicit derivation of the Σ(L) groups for all links of 8 and fewer crossings. If one is interested in classifying knots and links up to oriented, labeled ambient isotopy, it is important to know the symmetry group Σ(L) for each prime link type L, since links related by an element in Γ(L) outside Σ(L) are not (oriented, labeled) ambient isotopic. We treat this topic in a forthcoming manuscript [11]
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