Title: Topological Semimetals carrying Arbitrary Hopf Numbers: Hopf-Link, Solomon's-Knot, Trefoil-Knot and Other Semimetals,
Abstract: 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 $(3/2,1)$ 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.