Many things about direct products of knots are still unknown. The main unanswered question is still: if c(L) is the crossing number of a knot or link L, is it true that c(L1#L2) = c(L1)+c(L2)? This problem has been open for more then 100 years. From the Kauffman-Murasugi Theorem it follows that the conjecture holds when L1#L2 is an alternating KL (Kauffman, 1988). In a similar way, the old conjecture on unknotting (unlinking) numbers u(L1#L2) = u(L1)+u(L2) is still unproved. It is known that the relationship u(L1#L2) £ u(L1)+u(L2) holds, and M. Scharlemann (1985) proved that the conjecture is true in the case u(L1#L2) = 1. For u¥ numbers we conjecture that u¥ (L1#L2) = u¥ (L1)+u¥ (L2).
Given a projection of a KL, let us define an overpass to be a sub-arc of the KL that goes over at least one crossing, but never goes under a crossing. A maximal overpass is an overpass that could not be made any longer, so both of its endpoints occur just before we go under a crossing. The bridge number of a projection is the number of maximal overpasses in the projection. These maximal overpasses form the bridges over the rest of link L. The bridge number of L, denoted by b(L), is the least bridge number of all of the projections of L (see, e.g., Adams, 1994). Rational KLs are exactly 2-bridge KLs. For bridge numbers of composite KLs the equality b(L1#L2) = b(L1)+b(L2)-1 was proven by H. Schubert in 1954.
The LinKnot function fDToDDirect (webMathematica fDToDDirect) calculates the Dowker code of a direct product of KLs. The function fGenSignDirProd (webMathematica fGenSignDirProd) calculates the signs of a direct product of KLs in the order corresponding to the Dowker code or P-data.