Slavik JABLAN and RADMILA SAZDANOVIĆ |

AMPHICHEIRALITY
The research of amphicheirality started with works of J.B. Listing. He showed that the figure-eight knot, in honor of his accomplishment to knot theory called Listing knot is equal to its mirror image (this means, that figure-eight knot is amphicheiral), recognized that the left trefoil is different from the right trefoil knot (this means, that trefoil knot is chiral), and introduced a writhe of a knot, i.e., a signed sum of all crossings in a knot diagram.
The first results in the derivation of knots belong to P.G. Tait. The derivation of knots with 10 crossings took him six years to complete. In his works Tait also considered some of the fundamental problems in knot theory: chirality and unknotting number (or Gordian number, called by Tait "beknottedness"), and introduced the graph of a knot.
Thirty years after Tait's first results in enumeration of
achiral knots with at most crossings, M.G. Hasseman in his dissertation
partially extended knot tables, and described achiral knots with at most
12 crossings. Tait conjectured that every achiral knot or link must have
an even number of crossings. Therefore neither Tait nor Haseman considered
the possibility of the existence of achiral knots with an odd crossing
number. The first oriented achiral link 8
In the case of rational knots all amphicheiral knots are
derived from the same source: from the figure-eight knot 2 2. In general,
a knot or link is achiral (or amphichiral) if its "left" and "right"
forms are equivalent, meaning that one can be transformed to the other by
an ambient isotopy. If a knot
The n. Calculating the number of achiral
knots for n=2k (k=1, 2, 3,…) we obtain the Jacobsthal
sequence 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461,
10923, 21845, 43691, 87381, 174763, ...defined by the recursion
and given by the general formula
In recognizing other achiral knots we can use their
antisymmetrical rigid representations in
By using the function
KLs for which we need all non-isomorphic projections and to extend
that properties from individual projections to the corresponding KLs.
For example, if we find at least one achiral projection of a KL, we
know that the KL in question is achiral. This mainly holds for the
functions dealing with symmetry of KLs. In the same way as with
graph automorphisms, we can deal with projections of KLs as with
weighted graphs, where to every vertex is assigned its weight– a sign of
the vertex, and instead of classical automorphisms we consider
automorphisms preserving signs. As before, those automorphisms make a
group: the automorphism group of a KL.
The KL given by its Conway
symbol, Dowker code, or P-data, and the function
tests the chirality of an alternating knot or link given by its Conway
symbol. For example, the projection (((1,2,1,1),1,1,1),1,1) of the achiral
knot 2 3 3 2 is not achiral, but about achirality of that knot we can
conclude from the projection 2 3 3 2AmphiAltKL |