1.8.1 Chirality of rational KLs
Rational KLs are the main class of KLs for which we are able to analyze various general properties and construct large (infinite) subclasses of KLs satisfying those properties. Such a property is chirality: a KL is achiral (or amphicheiral) if its "left" and "right" forms are equivalent, meaning that one can be transformed to the other by an ambient isotopy (Liang and Mislow, 1994a, 1994b, 1995). We distinguish chirality of non-oriented and oriented KLs. An oriented KL is achiral if there is an ambient isotopy transforming the oriented link L into the oriented mirror image of the link L and preserving the orientation of components. For example, Hopf link 222 (2) is achiral as non-oriented and chiral as an oriented link. An oriented knot K is achiral iff it can be represented by an antisymmetrical vertex-bicolored graph on a sphere (Liang and Mislow, 1994b). In this case, for the oriented knot K there exist symmetry transposing orientations of vertices, i.e., mutually exchanging vertices with the signs +1 and -1. Its antisymmetries (this means sign-changing symmetries) could be sense-reversing- rotational antireflection and anti-inversion, or sense-preserving- 2-antirotation.
If the antisymmetry group
of the vertex-bicolored graph corresponding to an achiral oriented knot
contains at least one such sense-reversing antisymmetry, we have an invertible
achiral knot; otherwise, it is non-invertible. For the knot 2 2, the graph
symmetry group is G = [2+, 4], and the knot symmetry group G' = [2+, 4+]
is generated by the rotational reflection, with the axis defined by the
midpoints of colored (i.e., double) edges of the tetrahedron.
Considering the sign of the vertices, it is a rotational antireflection.
Its effect is preserved in all rational knots with an even number of crossings
that have a symmetrical (palindromic) Conway symbol. Hence, a rational
knot is achiral
iff its Conway symbol is symmetrical and has an
even number of crossings (Caudron, 1982). An oriented knot is invertible
if it remains the same after reversing its orientation. For
12, achiral rational knots are:
and all of them are invertible.