论文标题
交替结的周期性和自由周期性
Periodicity and Free Periodicity of Alternating Knots
论文作者
论文摘要
一个结$ k $ in $ s^3 $是$ q $ periodic，如果它承认与订单$ q $ of $ s^3 $的旋转相结合的对称性。如果$ k $承认对称性是同态的，没有定期的$ q $的$ s^3 $，则$ k $被称为自由$ q $ periodic。 在上一篇论文中，由于Menasco和thisLethwaite而乘飞机定理，我们获得了$ q $ Q $ Q> 2 $，可以在交替的投影中可视化作为投影球体的旋转。 在本文中，我们表明，交替结的免费$ q $ - 可以在某些交替投影上表示，作为订单$ q $旋转的组成，其中一些飞台都在其必需的Conway分解的相同扭曲的频带图上发生。因此，为了使交替结自由周期性，其基本分解必须满足某些条件。我们表明，任何免费或非免费$ q $ - action均以某种方式可见（几乎可见），并提供了一些足够的标准来检测几乎可见的预测中$ q $ actions的存在。 最后，我们展示了Murasugi分解为原子中的分解如何使我们能够确定自由$ Q $ Q $ - 周期的交替结的可见性$（q，r）$（$（q，q，r）$ - 镜头结）；实际上，我们只需要专注于其Murasugi分解的某种原子即可推断其可见性类型。
A knot $K$ in $S^3$ is $q$-periodic if it admits a symmetry that is conjugate to a rotation of order $q$ of $S^3$. If $K$ admits a symmetry which is a homeomorphism without fixed point of period $q$ of $S^3$, then $K$ is called freely $q$-periodic. In a previous paper, we obtained, as a consequence of Flyping Theorem due to Menasco and Thislethwaite, that the $q$-periodicity with $q>2$ can be visualized in an alternating projection as a rotation of the projection sphere. In this paper, we show that the free $q$-action 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 somehow visible (virtually visible) and give some sufficient criteria to detect the existence of $q$-actions from virtually visible projections. Finally, we show how the Murasugi decomposition into atoms enables us to determine the visibility type $(q,r)$ of the freely $q$-periodic alternating knots ($(q,r)$-lens knots); in fact, we only need to focus on a certain atom of their Murasugi decomposition to deduce their visibility type.
