2.9  Projections of KLs and chirality 

As we have already seen, a KL can have more then one different minimal projection. In the minimal projections we can denote chains of digons by bold lines. In the case of alternating KLs every minimal projection can be obtained from any other by a series of flypes. If every chain of digons is represented by a bold line, an elementary flype can be illustrated as in the given figure. If we collapse every chain of digons with both ends connected with the same vertex into a black point, and again denote new digons obtained after that collapse by bold lines, we will obtain vertex-bicolored diagrams of KLs, where a transition from one projection to the other by a flype can be represented by the mutual exchange of differently colored vertices connected by a bold edge. For example, the transfer from one minimal projection of the link 2 2 2 to the other projection ((1,2,1),1,1) of the same link expressed in the language of such vertex-bicolored flype diagrams is illustrated in the corresponding figure. This flyping algorithm works only for algebraic tangles. Non-isomorphic projections of KLs with n £ 9 crossings and their corresponding vertex-bicolored flype diagrams are given in the preceding figures. For a recognition of non-isomorphic alternating KL projections and their chirality, two additional polynomial projection invariants can be introduced. A KL is achiral iff it is equal to its mirror image. Because the achirality represents an equilibrium state with regards to "left" and "right" orientation, it is clear that for every achiral projection of an oriented KL the writhe will be equal to 0. This is a necessary (but not sufficient, and very weak) condition for the achirality of oriented KLs. First chiral alternating knot with the writhe equal to 0 is 4 1 3 with n=8 crossings. From 26 knots with n=10 crossings and with zero writhe, 14 are achiral, and 12 are chiral: 2 1 0:2:2 0, 2 2 1 1 1 1 2, 2 2,2 1,2+, 2 3,3,2, 3 0:2 0:2 0, 3 1 1,3,2, 3 1 1 3 2, .3.2 0.2, 3:2 0:2 0, .3.2.2 0, 4 1 1 1 3, and 4 1,3,2. As a necessary and sufficient condition we propose the following rule: an oriented KL is achiral iff one of its projections (onto the plane Â2 or on the sphere S3) can be divided into two mutually antisymmetrical parts, i.e. in two parts that can be transformed one to another by some vertex sign-changing transformation.

Let us consider an oriented alternating KL diagram D with generators g1,... ,gn. In every vertex of D there is a passing generator gi, and incoming and outgoing generators gj, gk, respectively. If e(V) is the sign of the crossing V, then aii=e(V) t, aij=1, aik=-1, and d(t)= det(aij). For example, for the achiral figure-eight knot 2 2 (or 41):