# Nodal-knot semimetals

## TL;DR

This paper introduces a new class of topological semimetals featuring knotted nodal lines in the Brillouin zone, connecting knot theory with topological materials and exploring their transitions.

## Contribution

It presents a generic construction method for nodal-knot semimetals, including trefoil and more complex knots, and studies their topological transitions.

## Key findings

- Introduced a construction for nodal-knot semimetals
- Demonstrated trefoil and complex nodal knots in Brillouin zone
- Analyzed topological transitions via nodal-line reconnections

## Abstract

Topological nodal-line semimetals are characterized by one-dimensional lines of band crossing in the Brillouin zone. In contrast to nodal points, nodal lines can be in topologically nontrivial configurations. In this paper, we study the simplest topologically nontrivial forms of nodal line, namely, a single nodal line taking the shape of a knot in the Brillouin zone. We introduce a generic construction for various "nodal-knot semimetals", which yields the simplest trefoil nodal knot and other more complicated nodal knots in the Brillouin zone. The knotted-unknotted transitions by nodal-line reconnections are also studied. Our work brings the knot theory to the subject of topological semimetals.