Still there is no simple criterion for the recognition of achiral knots, even for alternating ones. One of the attempts was Kauffman Conjecture (Kauffman, 1990a; van Mill and Reed, 1991):

Conjecture Let K be an achiral alternating knot. Then there exists a reduced alternating diagram D of K, such that G(D) is isomorphic to G*(D), where G(D) is a checkerboard-graph of D and G*(D) its dual.

Its counterexample, the knot (2 1,3) 1 1 (3,2 1), was found by Dasbach and Hougardy (1996). They proved that none of the eight graphs corresponding to different embeddings of this knot is
isomorphic to its dual.

The same knot can be used as the example that it is not an easy task to find an antisymmetrical drawing that shows concealed antisymmetry, e.g., the achirality of the projection (2 1,3) 1 1 (3,2 1). Its achirality can be explained by the presence of the rotational antireflection, i.e., by purely geometrical antisymmetry arguments. As we concluded many times before for different KL properties, achirality is a property of families. For example, from the knot (2 1,3) 1 1 (3,2 1) we derive an infinite family of achiral knots ((2m) 1,(2n+1)) 1 1 ((2n+1),(2m) 1), (m,n ³ 1).

In the same way as with achiral alternating knots, we can work with achiral alternating oriented links. The first of them is the basic polyhedron 6^*- Borromean rings. For n=8 crossings we will have the stellar achiral 4-component link 2,2,2,2 that generates two infinite families of 4-component achiral links: (2m),(2m),(2n),(2n) and (2m),(2n),(2m),(2n), (m,n ³ 1); the arborescent achiral 3-component link (2,2) (2,2) that generates the infinite family of achiral 3-component links ((2m),(2n)) ((2m),(2n)), (m,n ³ 1); and the polyhedral 3-component link .2:2 0 that generates the infinite family of 3-component achiral links .(2m):(2m) 0 (m ³ 1). For n=10 we have the beginnings of four new infinite families of achiral links: the 3-component achiral link (2,2+) (2,2+) that generates the infinite family of 3-component achiral links ((2m),(2n)+(2p)) ((2m),(2n)+2p), (m,n,p ³ 1); the 3-component achiral link .3.3 that generates the infinite family of achiral 3-component links .(2m+1).(2m+1), (m ³ 1); the 3-component achiral link .3:3 0 that generates the infinite family of 3-component achiral links .(2m+1):(2m+1) 0; and the 3-component achiral link 8^*2.2 that generates the infinite family of achiral 3-component links 8^*(2m+1).(2m+1), (m ³ 1). For n=12, together with the links belonging to the families mentioned (4,4,2,2; 4,2,4,2; (4,2) (4,2); (2,2++) (2,2++), and 4.:4 0), we have new beginnings of infinite families of achiral links: