1.14 More KL invariants
In the case of rational KLs it was not a problem to reduce them and obtain minimal projections: we have been able to use a very fast and simple function RatReduce based on the algorithm containing only two Mathematica functions: ContinuedFraction and FromContinued Fraction, applying them successively to a KL given in the Conway notation. To recognize and compare KLs from other KL-worlds, we need more refined invariants and reduction methods. For all alternating KLs, one of them is a minimization of Dowker codes. We already mentioned Tait's Flyping Theorem. According to it we can switch between any two of alternating KL projections by a sequence of flypes.
In the language of the Conway notation, a product p q can be expressed as a ramification (p,1,...,1), where 1 occurs q times. The flyping sequence for a product p q is then: (p,1,...,1), (1,p, ...,1), ..., (1,1,...,p). Using the same flyping algorithm for Conway symbols, we can obtain all projections of a KL given by its Conway symbol. The LinKnot function fProjections (webMathematica fProjections), written with the kind help of M. Sremcevic, calculates all projections an alternating KL given in the Conway notation, their Conway symbols and the overall number. From the obtained projections we can select non-isomorphic ones (i.e., their representatives) by using the function fDiffProjectionsAltKL (webMathematica fDiffProjectionsAltKL). Then, from each projection we can obtain its Dowker code by applying to it the function Dow, but the codes obtained will be not necessarily minimal. To minimize them, we can use the function MinDowProjAltKL (webMathematica MinDowProjAltKL). Finally, the LinKnot function MinDowAltKL (webMathematica MinDowAltKL) produces a minimal Dowker code for any alternating KL. Two such alternating prime KLs are equal (up to their mirror images) iff their minimal Dowker codes without signs are equal. This means that minimal Dowker codes (even without signs) are sufficient to recognize the equality of any two alternating knots or links L1 and L2 given by their Conway symbols. The function SameAltProjKL (webMathematica SameAltProjKL) compares two alternating KL projections given by their Conway symbols. The result is 1 for equal (that is, isomorphic), and 0 for non-equal projections. The function SameAltConKL (webMathematica SameAltConKL) compares two alternating KLs given by their Conway symbols. The result is 1 for equal (that is, ambient isotopic), and 0 for non-equal KLs. Important remark: in the case of polyhedral KLs derived from basic polyhedra that permit flypes only the function MinDowProjAltKl works correctly, because the flypping algorithm used in the program LinKnot does not work with non-algebraic tangles!
By using the function fDiffProjectionsAltKL we are able to calculate some invariants (or properties) of alternating KLs for which we need all non-isomorphic projections and to extend those properties from individual projections to the corresponding KL. For example, if we find at least one achiral projection of a KL, we know that the KL in question is achiral. This mainly holds for the functions dealing with symmetry of KLs. In the same way as with graph automorphisms, we can deal with KL projections as with weighted graphs, where to every vertex is assigned its weight- a sign of the vertex, and instead of classical automorphisms we consider automorphisms preserving signs. As before, those automorphisms make a group: the automorphism group of a KL. The order of the automorphism group Aut(L) of a link L is the number of (sign-preserving) automorphisms it contains.