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

## TL;DR

This paper introduces a new class of topological semimetals characterized by arbitrary Hopf numbers, with complex linked Fermi surfaces such as knots and links, expanding the understanding of topological phases.

## Contribution

It proposes a novel framework for Hopf semimetals with tunable Hopf numbers, including rational values, and describes their linked Fermi surface structures.

## Key findings

- Fermi surfaces form torus links like Hopf links, Solomon's knots, and trefoil knots.
- Hopf number can be arbitrarily chosen as an integer or rational.
- Fermi surfaces can be open strings for rational Hopf numbers.

## 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.