2.8.1 Nonalternating and almost alternating KLsThe list of nonalternating 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 a_{1},...,a_{n}+ to a_{1},...,a_{n}, and 1^{} denotes the crossing change from +1 to 1.
Generally, the problem of finding minimal
almostalternating representations is a difficult problem with a lot of open
questions. For some almost alternating KLrepresentations, their First we need to compare the existing results for the number of nonalternating knots. For n £ 9 all the sources (Conway, 1970; Caudron, 1982; Adams, 1994; Rolfsen, 1976) are fully consistent: there are 3 nonalternating 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. 10_{162}=10_{161}, was repeated (Perko pair). After that correction for n=10 there are 42 nonalternating knots. For n=11 in Conway's paper we find 182 nonalternating 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 nonalternating links the sources (Conway, 1970; Caudron, 1982; Rolfsen, 1976; Doll and Hoste, 1991) coincide with regard to nonalternating links with n£ 9 crossings: there is one 3component link for n=6, two 2component links for n=7, 8 nonalternating links for n=8 (two 2component, four 3component and two 4component links), and 28 nonalternating links for n=9 (nineteen 2component and nine 3component 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 nonalternating links with n=10 crossings, recently completed by M. Thistlethwaite, who obtained 116 nonalternating 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 nonalternating knots in the list of 11crossing 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 nonalternating KLs, the general rules for their derivation can be introduced in the case of stellar and arborescent nonalternating KLs. The same holds for almost alternating representations of stellar and arborescent KLs. All stellar nonalternating KLs of the form a_{1},...,a_{i}, i=3,4,... can be directly derived from the KLs of the form a_{1},...,a_{i}+ if we replace + by . In the same way, nonalternating arborescent KLs of the form (a_{1},a_{2}) (a_{3},a_{4}) can be expressed by the almost alternating minimal representation (a_{1},a_{2}) (a_{3},a_{4}+^{}), etc. In this way, almost alternating representations of some nonalternating polyhedral KLs can also be obtained. The remaining nonalternating KLs that cannot be derived in such a direct way are given in the following tables.
