Non-Hermitian eigenvalue knots and their isotopic equivalence

TL;DR

This paper explores the topology of eigenvalue knots in non-Hermitian systems, demonstrating their construction, isotopic equivalence, and relation to spectral topology through experiments and topological analysis.

Contribution

It introduces a method to construct eigenvalue knots, analyzes their isotopic relations, and links spectral topology with knot theory in non-Hermitian systems.

Findings

Experimental realization of knots with braid index 3

Eigenvalue knots are isotopic under homotopic parametric loops

Eigenvalue knots contain information beyond spectral topology

Abstract

The spectrum of a non-Hermitian system generically forms a two-dimensional complex Riemannian manifold with distinct topology from the underlying parameter space. Spectral topology permits parametric loops to map the affiliated eigenvalue trajectories into knots. In this work, through analyzing exceptional points and their topology, we uncover the necessary considerations for constructing eigenvalue knots and establish their relation to spectral topology. Using an acoustic system with two periodic synthetic dimensions, we experimentally realize several knots with braid index 3. In addition, by highlighting the role of branch cuts on the eigenvalue manifolds, we show that eigenvalue knots produced by homotopic parametric loops are isotopic such that they can deform into one another by type-II or type-III Reidemeister moves. Our results not only provide a general recipe for constructing eigenvalue knots but also expand the current understanding of eigenvalue knots by showing that they contain information beyond that of the spectral topology.