Every 4-valent graph induces a corresponding alternating KL diagram. The program LinKnot contains the database of basic polyhedra with at most n = 20 crossings, where every basic polyhedron is represented by its corresponding alternating KL diagram. For the basic polyhedra with n < 10 crossings, the standard notation is used (.1, 6^*, 8^*, 9^*, where the symbol for 6^* can be omitted in symbols of KLs beginning with a dot). For example, the knot 10₉₅ is denoted by ".2 1 0.2.2", and 10₁₀₁ by "2 1..2..2". For n ³ 10 in each symbol the first two digits represent the number of crossings, and the next the ordering number of the polyhedron (e.g., 101^*, 102^*, 103^* for n = 10 denoting 10^*, 10^**, 10^***, respectively, and 111^*, 112^*, 113^* for n = 11 denoting 11^*, 11^**, 11^***, respectively, etc.). For n = 12 basic polyhedra are ordered according to their list made by A. Caudron (1982), so polyhedra originally denoted with 12A-12L are 121^*-1212^*. For n > 12 the database of basic polyhedra is produced from the list of simple 4-regular, 4-edge-connected, but not necessarily 3-vertex connected plane graphs generated by Brendan McKay using the program plantri written by Gunnar Brinkmann and Brendan McKay. You can work directly with the basic polyhedra up to 16 crossings from the file PolyBase.m that downloads automatically. For n = 12 it contains 12 basic polyhedra beginning from 121^* to 1212^*, for n = 13 it contains 19 basic polyhedra from 131^* to 1319^*, for n = 14 it contains 64 basic polyhedra from 141^* to 1464^*, for n = 15 this file contains 155 basic polyhedra from 151^* to 15155^*, and for n = 16 it contains 510 basic polyhedra, beginning from 161^* to 16510^*. In order to work with the basic polyhedra from n = 17 to n = 20 vertices, you need to open an additional database PolyBaseN.m, for n = 17 to n = 20 (by writing, e.g. << PolyBase17.m or Needs["PolyBase17.m"] for n = 17, etc.). They contain, respectively, 1514 basic polyhedra for n = 17 from 171^* to 171514^*, 5146 basic polyhedra for n = 18 from 181^* to 185146^*, 16966 basic polyhedra for n = 19 from 191^* to 1916966^*, and 58782 basic polyhedra for n = 20 from 201^* to 2058782^*. In the symbols of polyhedral KLs omitted vertices are denoted by a series of single dots (without using colons as :, :. or ::), and with zeros separated by a space. For example, the knot 10₁₅₆ is denoted by "-3..2..2  0" (with a series of single dots and 0 separated by a space). 

For n £ 11 the correspondence between basic polyhedra (source links) and their corresponding alternating KLs is one-to-one: every alternating KL corresponding to a basic polyhedron (source link) has exactly one minimal projection. However, for n = 12 the 4-component link 11^***2 gives as its other shadow the basic polyhedron 125^* (or 12E according to Caudron's list), and for n = 13 the 4-component link gives as its minimal projections the basic polyhedra 136^* and 1318^*. In the both cases, one projection can be obtained from the other by a single flype, so for the corresponding alternating links we have the equalities 11^***2 ~ 125^* and 136^* ~ 1318^*, where the symbol ~ is used to denote this equality. Hence, for n ³ 12 it is necessary to make distinction between basic polyhedra or source links (treated as KL shadows, i.e., 4-valent graphs) and their corresponding alternating KLs.