The LinKnot function AmphiProjAltKL (webMathematica AmphiProjAltKL) tests chirality of a given projection of an oriented alternating KL given by its Conway symbol, Dowker code, or P-data, the function AmphiAltKL (webMathematica AmphiAltKL) tests the chirality of an alternating oriented KL given by its Conway symbol, and the function AmphiQ (webMathematica AmphiQ) tests chirality of non-oriented KL by comparing Kauffman polynomials of a KL and its mirror image. The first two functions are based on antisymmetrical representations of achiral projections. For example, the projection (2 2,2) (2 2,2) of the achiral knot (2 2,2) (2 2,2) is not achiral, but about achirality of that knot we can conclude from the projection ((1,1,2),2) ((2,1,1),2). The next figure shows its centro-antisymmetrical representation. Because those functions are able to recognize as achiral only alternating KLs having at least one antisymmetrical minimal projection, they fail in the case of achiral oriented KLs without antisymmetrical minimal projections (e.g., for the 3-component links 3.2.3 0:2, 8*2.2:2 0:.2 0, 8*2 0.2 0:.2 0.2 0, and 10**.2::.2 with n = 12 crossings, for the knot 10***2:2:.2 0:2 0.2 1.2 1 0 with n = 18 crossings, etc.). 

If we can rotate a KL projection by an angle [(2p)/p] about a certain axis so that it rotates to its original shape, we say that this projection has period p. Even a single projection of a KL can have several different periods. For example, there is only one projection of a trefoil knot 3. For a rotation axis we have two possible choices, the first corresponding to three-fold, and the second to two-fold rotation (a half-turn). Hence, the periods of a trefoil projection are 3 and 2. For a KL with several non-isomorphic projections, we can compute periods for all of them. The list of periods of a KL consists of all possible periods of its projections. Certainly, the number of all possible projections of a KL is infinite, so we are working only with alternating KLs and all their minimal projections. The LinKnot function PeriodProjAltKL (webMathematica PeriodProjAltKL) calculates periods of a given projection of an alternating KL given by its Conway symbol, Dowker code, or P-data, and the function PeriodAltKL (webMathematica PeriodAltKL) calculates the period of a given alternating KL given by its Conway symbol. Both functions give complete results (all possible periods) for most of KLs. 


PreviousContentsNext