Periodicity and free periodicity of alternating knots

Abstract

In a previous paper [6], we obtained, as a consequence of Flyping Theorem due to Menasco and Thislethwaite, that the q-periodicity (q>2) of an alternating knot can be visualized in an alternating projection as a rotation of the projection sphere. See also [2].In this paper, we show that the free q-periodicity (q>2) of an alternating knot can be represented on some alternating projection as a composition of a rotation of order q with some flypes all occurring on the same twisted band diagram of its essential Conway decomposition. Therefore, for an alternating knot to be freely periodic, its essential decomposition must satisfy certain conditions. We show that any free or non-free q-action is some way visible (virtually visible) and give some sufficient criteria to determine from virtually visible projections the existence of a q-action.Finally, we show how the Murasugi decomposition into atoms as initiated in [12] and [13] enables us to determine the visibility type (q,r) of the freely q-periodic alternating knots ((q,r)-lens knots [3]); in fact, we only need to focus on a certain atom of their Murasugi decomposition to deduce their visibility type.