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. |