In Redmeister book [13] only knots are discussed, and they denoted by 3_{1}, 4_{1}, 5_{1}, 5_{2}, etc. In the similar way are composed 2-component link tables, 3-component link tables, etc. This classical notation, giving no information about any knot or link (except its place in knot tables), is preserved till now in the most of the knot theory books. |
In Dowker notation [11] every alternating knot is given by its minimal Dowker sequence (e.g. 4 6 8 2), from which is possible to reconstruct the knot. Because Dowker code is dependent from the minimal projection and from the choice of beginning point, the opposite way is very complicated. Dowker codes are just minimal permutations representing certain knots, without carying any other information about them, so they are absolutely unuseful in any attempt of knot and link classification. |
With the developement of computers it is possible to analyze links (by the program SnapPea), to construct all possible permutations of n even numbers, check their realizability as knot projections, and find the minimal Dowker sequence for every knot. This way M.B.Thistlethwaite (by the program Knotscape), J.Hoste [17] et all. obtained the tables of knots with n £ 16 crossings or nonisomorphic minimal link projections with n £ 9 crossings. For example, for n = 8 we have the following Dowker sequences for alternating knots: 4 8 10 14 2 16 6 12 4 8 12 2 14 6 16 10 4 8 12 2 14 16 6 10 4 8 12 2 16 14 6 10 4 8 14 10 2 16 6 12 4 10 12 14 2 16 6 8 4 10 12 14 2 16 8 6 4 10 12 14 16 2 6 8 4 10 12 14 16 2 8 6 4 10 14 16 12 2 8 6 4 10 16 14 12 2 8 6 6 8 10 12 14 16 2 4 6 8 12 2 14 16 4 10 6 8 12 14 4 16 2 10 6 8 14 12 4 16 2 10 6 10 12 14 16 4 2 8 6 10 12 16 14 4 2 8 6 12 10 16 14 4 2 8 |
The symbolical notation for knots and links was introduced by J.Conway in 1967 [14]. |