2.8.1  Non-alternating and almost alternating KLs

The list of non-alternating KLs with n£  11 crossings is given by J.H. Conway (1970) and A. Caudron (1982). In The Knot Book (1994) C.C. Adams introduced the idea of almost alternating KLs. A projection of a KL is called an almost alternating projection if one crossing change in the projection would make it an alternating projection. We call a KL an almost alternating KL if it has an almost alternating projection, and if it does not have an alternating projection. For all KLs with n£  11 crossings, with the exception of four of them, we were able to find their explicit minimal almost alternating representations. In their corresponding Conway symbols +- denotes the crossing change from a1,...,an+ to a1,...,an-, and 1- denotes the crossing change from +1 to -1.

Generally, the problem of finding minimal almost-alternating representations is a difficult problem with a lot of open questions. For some almost alternating KL-representations, their
minimality can be proved by the graph-theoretical argumentation (
Caudron, 1982), but for the most of them the only method still remains the empirical one: construct all almost alternating representations with a fixed number of crossings and choose the first corresponding to a given non-alternating KL. Whenever possible, we used the other approach: the request that the minimal not-alternating representation of a given KL has to belong to the same family (class, subworld,...) as the KL it represents.

First we need to compare the existing results for the number of non-alternating knots. For n £  9 all the sources (Conway, 1970; Caudron, 1982; Adams, 1994; Rolfsen, 1976) are fully consistent: there are 3 non-alternating knots for n=8 and 8 for n=9. Computer derivation done with the program Knotscape confirmed these results. For n=10 in Conway (1970) and Rolfsen (1976) one knot, 2 1:-2 0:-2 0=3:-2 0:-2 0, i.e. 10162=10161, was repeated (Perko pair). After that correction for n=10 there are 42 non-alternating knots. For n=11 in Conway's paper we find 182 non-alternating knots, where the knots -2 1 0:3:2, 8*-2 1 0:.2 and 8*-3 0::2 0 are missing. Their corrected list, containing 185 knots, is given by A. Caudron and confirmed by the computer derivation.

For non-alternating links the sources (Conway, 1970; Caudron, 1982; Rolfsen, 1976; Doll and Hoste, 1991) coincide with regard to non-alternating links with n£  9 crossings: there is one 3-component link for n=6, two 2-component links for n=7, 8 non-alternating links for n=8 (two 2-component, four 3-component and two 4-component links), and 28 non-alternating links for n=9 (nineteen 2-component and nine 3-component links). These results are confirmed by the computer derivation (Doll and Hoste, 1991). For n=10 the only sources are (Conway, 1970; Caudron, 1982). They need to be corrected according to the computer derived list of non-alternating links with n=10 crossings, recently completed by M. Thistlethwaite, who obtained 116 non-alternating links.

Since an algebraic link that has exactly one negative sign in its Conway notation has an almost alternating projection (Adams, 1994), that all but 18 of the non-alternating knots in the list of 11-crossing prime knots given in the Conway notation are almost alternating, and for those 18 we will try to find their almost alternating minimal representations.

As we mentioned before, in discussing the derivation of non-alternating KLs, the general rules for their derivation can be introduced in the case of stellar and arborescent non-alternating KLs. The same holds for almost alternating representations of stellar and arborescent KLs. All stellar non-alternating KLs of the form a1,...,ai-, i=3,4,... can be directly derived from the KLs of the form a1,...,ai+ if we replace + by -. In the same way, non-alternating arborescent KLs of the form (a1,a2) (a3,a4-) can be expressed by the almost alternating minimal representation (a1,a2) (a3,a4+-), etc. In this way, almost alternating representations of some non-alternating polyhedral KLs can also be obtained. The remaining non-alternating KLs that cannot be derived in such a direct way are given in the following tables.

8103 (2,2)-(2,2)  (2,2)(3 1-,2)          834 2,2,2,2- -  2,2,2,3 1-