DISCOVERING SYMMETRY OF KNOTS
BY USING PROGRAM LinKnot

Slavik JABLAN and RADMILA SAZDANOVIĆ

3 PERIODICITY

If we can rotate a knot or link projection by an angle 2π/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 knot or link 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 knot or link with several non-isomorphic projections, we can compute periods for all of them. The list of periods of a knot or link consists of all possible periods of its projections. Certainly, the number of all possible projections of a knot or link is infinite, so we are working only with alternating knots and links and all their minimal projections. The LinKnot function PeriodProjAltKL calculates periods of a given projection of an alternating knot or link given by its Conway symbol, Dowker code, or P-data, and the function PeriodAltKL calculates the period of a given alternating knot or link given by its Conway symbol.