Topological semimetals carrying arbitrary Hopf numbers: Fermi surface topologies of a Hopf link, Solomon's knot, trefoil knot, and other linked nodal varieties

We propose a type of Hopf semimetal indexed by a pair of numbers (p,q), where the Hopf number is given by pq. The Fermi surface is given by a 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 a Hopf link indexed by (1,1), Solomon's knot (2,1), a double Hopf link (2,2), and a double trefoil knot (3,2). We may choose p or q to be a half integer, where the Fermi surface is a torus knot, such as a trefoil knot (3/2,1). It is even possible to make the Hopf number an arbitrary rational number, where a semimetal whose Fermi surface forms open strings is generated.