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 is 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 the double trefoil-knot $(3,2)$. We may choose $p$ or $q$ as a half integer, where torus-knot Fermi surfaces such as the trefoil knot $(3\mathrm{/}2,1)$ are realized. It is even possible to make the Hopf number an arbitrary rational number, where a semimetal whose Fermi surface forms open strings is generated.