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Let us notice that all the families of P-undetectable KLs obtained from the antiprismatic basic polyhedra (2k)* (k=3,4,5,¼) are of the form (2k)*w.t1 x 0.w¢.t2 y 0, where w and w¢ are palindromic words of the form a.b¼ with the antisymmetrical distribution of signs. For the basic polyhedron 1449* that is isomorphic to 14* we can get the same conclusion after reordering vertices. From the basic polyhedron 10*** are obtained the following P-undetectable families:
Conjecture Multi-parameter families of P-undetectable KLs can be obtained from every achiral basic polyhedron. Conjecture All algebraic alternating KL families are detectable by any polynomial invariant P. Algebraic KL families can be only Alexander- and Conway-undetectable. Achiral basic polyhedra with n=12 crossings are: 12A (121*), 12B (122*), 12J (1210*), 12K (1211*), and 12L (1212*). In a search for Jones unknot, we need to work with Jones-undetectable families with t1=t2, and then try to find among them Jones unknot by varying the tangle t1 and other parameters.
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