“Slicing” the Hopf link

Links not concordant to the Hopf link

We give new Casson–Gordon style obstructions for a two–component link to be topologically concordant to the Hopf link.

A two component link with Alexander polynomial one is concordant to the Hopf link

Let L be a two component link in S, an embedding of two disjoint circles which is topologically locally flat, that is, which extends to an embedding of two solid tori. The link has Alexander polynomial one, if, and only if, the first homology of the universal abelian cover of the complement of the link vanishes. If the link has Alexander polynomial one, then the linking number of the two compon...

Survivable Probability of Network Slicing with Random Physical Link Failure

The fifth generation of communication technology (5G) revolutionizes mobile networks and the associated ecosystems through the integration of cross-domain networks. Network slicing is an enabling technology for 5G as it provides dynamic, on-demand, and reliable logical network slices (i.e., network services) over a common physical network/infrastructure. Since a network slice is subject to fail...

Topological Semimetals carrying Arbitrary Hopf Numbers: Hopf-Link, Solomon’s-Knot, Trefoil-Knot and Other Semimetals

We propose a new type of Hopf semimetals indexed by a pair of numbers (p, q), where the Hopf number is given by pq. The Fermi surface is given by the preimage of the Hopf map, which consists of loops nontrivially linked for a nonzero Hopf number. The Fermi surface forms a torus link, whose examples are the Hopf link indexed by (1, 1), the Solomon’s knot (2, 1), the double Hopf-link (2, 2) and t...

On the Coderivations of Some Hopf Algebras