In order to derive all composite basic polyhedra with n £ 15 crossings, 21 hyperbolic tangles are needed: one with n=5, one with n=7, two with n=8, six with n=9, and eleven with n=10 crossings. Non-algebraic tangles with n=11 crossings are given in the corresponding figure. Among basic polyhedra with n=14 crossings, there are 27 composite polyhedra, 18 of their corresponding KLs permit flypes, and among them 15 will have more then one minimal projection. Links associated to the basic polyhedra with n=14 crossings satisfy following equalities : 1429* ~ 1430*, 1434* ~ 1445*, 1435*~ 1439*, 1455*~ 1456*~ 1458*, 1463* ~1464*, and the links 1459* - 1464* have other projections that contain bigons. Among 76 composite basic polyhedra with n=15 crossings, 59 of the corresponding KLs permit flypes, and among 257 composite basic polyhedra with n=16 crossings, 201 of the corresponding KLs permit flypes.
Starting with composite basic polyhedra representations we can make interesting conclusions about the properties of their corresponding alternating KLs. For example, KLs corresponding to the "palindromic" basic polyhedra 10***=5* 5*, 12J=5* 1 1 5* , 1420*=7** 7**, 16160*=7** 1 1 7** are achiral from the same reason as their analogous rational KLs from the class p p, where p is an arbitrary tangle. The same holds for all "palindromic" basic polyhedra of the form p' p', where p' is any non-algebraic tangle.
Basic polyhedra representations are not unique: the same composite basic polyhedron (e.g., 16268*) can be represented by different hyperbolic tangle decompositions. As we have seen, for n ³ 13 two or more different basic polyhedra (e.g., 1318* and 136*) can be obtained as non-isomorphic projections of the same alternating KL, so even the term "basic polyhedron of an alternating KL" must be reconsidered. For example, the Conway symbols 125*2 0 and 11***2 1 represent the same alternating link. In the same way, 136*2=1318*2, etc. The same holds for source KLs with n ³ 12 crossings: two or more mutually non-isomorphic source links can be projections of the same alternating KL. For example, source links derived from the basic polyhedron 11***=5* 1 5* , like two non-isomorphic source links 11***.2 and 11***::2 that are two different projections of the same alternating source link. The same property holds for source links derived from the basic polyhedra 12E, 12J... In order to avoid possible ambiguity, instead of defining a basic polyhedron and source link as a graph, it is possible to define it as an alternating KL corresponding to this graph and introduce extended Conway notation for composite basic polyhedra.
The LinKnot function fCompositePoly (webMathematica fCompositePoly) detects composite basic polyhedra, and the function fPolyFlype (webMathematica fPolyFlype) finds those with the corresponding KLs that permit flypes.
The LinKnot functions fProdTangles (webMathematica fProdTangles) and fSumTangles (webMathematica fSumTangles) calculate P-data of the product and sum of non-algebraic tangles (denoted by m5*, m7*, m81*-m82*, m91*-m96*, m101*-m1011*, m111*-m1138*) composed with algebraic tangles placed in the basic polyhedron obtained by the product or sum.