It is well known that every orientation preserving homeomorphism of the plane $\mathbb{R}^2$ with a $k$-periodic orbit ${\cal O}$ ($k \geq 2$) necessarily has a fixed point (see [4] or [5], [7], [11]). It is then interesting to know if ${\cal O}$ is linked with one of these fixed points. The difficulty in answering this question depends a lot on the precise sense we give to the word “linked”. Before discussing briefly this point, let us observe this general problem has a very natural counterpart in the framework of orientation reversing homeomorphisms of the sphere $\mathbb{S}^2$. Indeed, every such homeomorphism possessing a $k$-periodic orbit ${\cal O}$ ($k\geq 3$) also has a 2-periodic orbit ([2]) and one can ask if ${\cal O}$ is linked with one of these 2-periodic orbits. The aim of this paper is to try to answer this question.