# Hyperbolic Nodal Band Structures and Knot Invariants

## TL;DR

This paper explores hyperbolic knotted nodal structures in band topologies, demonstrating their stability in both Hermitian and non-Hermitian systems, and introduces an efficient algorithm for their topological classification.

## Contribution

It extends the classification of band structures to include hyperbolic knots and links, providing new insights into their stability and topological invariants.

## Key findings

- Hyperbolic nodal links and knots are present in both Hermitian and non-Hermitian systems.
- The stability of these structures depends on discrete symmetries in Hermitian systems and is inherently robust in non-Hermitian systems.
- An efficient algorithm for computing Alexander polynomial, linking numbers, and Milnor invariants was developed.

## Abstract

We extend the list of known band structure topologies to include a large family of hyperbolic nodal links and knots, occurring both in conventional Hermitian systems where their stability relies on discrete symmetries, and in the dissipative non-Hermitian realm where the knotted nodal lines are generic and thus stable towards any small perturbation. We show that these nodal structures, taking the forms of Turk's head knots, appear in both continuum- and lattice models with relatively short-ranged hopping that is within experimental reach. To determine the topology of the nodal structures, we devise an efficient algorithm for computing the Alexander polynomial, linking numbers and higher order Milnor invariants based on an approximate and well controlled parameterisation of the knot.