Generating Exceptional Knots and Links with Arbitrary Braiding Topology

TL;DR

This paper introduces a universal framework for creating and classifying knotted and linked exceptional loops in 3D non-Hermitian systems, linking knot theory with topological phases and demonstrating experimental feasibility.

Contribution

It provides a systematic method to realize arbitrary knotted exceptional loops in non-Hermitian Hamiltonians using braid representations, establishing a topological classification based on knot invariants.

Findings

Constructed minimal two-band Hamiltonians for prescribed knot topologies.

Demonstrated stability and topological transitions of knotted exceptional loops.

Proposed experimental realization in electro-acoustic systems.

Abstract

Non-Hermitian systems host band degeneracies that are fundamentally distinct from those in Hermitian systems, most notably exceptional points (EPs) where both eigenvalues and eigenvectors coalesce. In three dimensional (3D) non-Hermitian systems, such degeneracies can form closed exceptional loops (ELs), whose global geometry can exhibit nontrivial knot and link structures. In this work, we present a universal and constructive framework for realizing knotted and linked ELs in 3D systems, establishing a direct correspondence between knot theory and non-Hermitian band degeneracies. Starting from an arbitrary knot or link specified by a braid representation, we systematically construct minimal two-band non-Hermitian Hamiltonians whose ELs faithfully realize the prescribed topology in momentum space, enabling a classification of non-Hermitian topological phases based on knot invariants such as braid words and Alexander polynomials. We show that these knotted ELs are generically stable and give rise to non-Hermitian metallic phases characterized by Seifert surfaces, reflecting the defective nature of exceptional degeneracies, in sharp contrast to nodal lines in Hermitian systems that typically require symmetry protection or fine-tuning. Furthermore, we demonstrate that knotted ELs can be continuously deformed and untied through controlled topological transitions driven by a single tuning parameter, providing a deterministic mechanism for manipulating knot topology in momentum space. We also propose an experimental realization in electro-acoustic systems, demonstrating the feasibility of observing knotted ELs through nonreciprocal coupling and tunable parameters. Our results establish knot and link topology as a natural classification scheme for non-Hermitian topological matter and suggest broad applicability in engineered platforms such as photonic, acoustic, and circuit-based systems.