Search results for "Hopf link"
showing 3 items of 3 documents
THE ZONE MODULUS OF A LINK
2005
In this paper, we construct a conformally invariant functional for two-component links called the zone modulus of the link. Its main property is to give a sufficient condition for a link to be split. The zone modulus is a positive number, and its lower bound is 1. To construct a link with modulus arbitrarily close to 1, it is sufficient to consider two small disjoint spheres each one far from the other and then to construct a link by taking a circle enclosed in each sphere. Such a link is a split link. The situation is different when the link is non-split: we will prove that the modulus of a non-split link is greater than [Formula: see text]. This value of the modulus is realized by a spec…
Bifurcations of Links of Periodic Orbits in Mathieu Systems
2000
We prove that orbits escape from infinity, and that therefore the sphere S can be considered as its phase space. If the parameter δ is large enough, the system is non-singular MorseSmale, and its periodic orbits define a Hopf link. As δ decreases, the system undergoes some bifurcations that we describe geometrically. We relate the bifurcation orbits to periodic orbits continued from the linear Mathieu equation.
Heisenberg quasiregular ellipticity
2016
Following the Euclidean results of Varopoulos and Pankka--Rajala, we provide a necessary topological condition for a sub-Riemannian 3-manifold $M$ to admit a nonconstant quasiregular mapping from the sub-Riemannian Heisenberg group $\mathbb{H}$. As an application, we show that a link complement $S^3\backslash L$ has a sub-Riemannian metric admitting such a mapping only if $L$ is empty, the unknot or Hopf link. In the converse direction, if $L$ is empty, a specific unknot or Hopf link, we construct a quasiregular mapping from $\mathbb{H}$ to $S^3\backslash L$. The main result is obtained by translating a growth condition on $\pi_1(M)$ into the existence of a supersolution to the $4$-harmonic…