Generating Exceptional Knots and Links with Arbitrary Braiding Topology

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.

• The paper establishes a universal four-step method that converts braid words into semiholomorphic polynomials and explicit two-band tight-binding Hamiltonians realizing arbitrary knot and link topologies as exceptional loops.
• The construction reproduces examples including trefoils, figure-eight knots, Hopf and Borromean links, with polynomial degree determining hopping range and crossing signs encoded directly in Hamiltonian parameters.
• The authors demonstrate continuous untying through a single scaling parameter, including reconnection events and exceptional chains, and propose electro-acoustic feedback as an experimental platform.

Overview

The paper by Jiang, Guo, Cai, and Jiang (2603.04143) develops a universal, constructive framework for realizing exceptional loops (ELs) with arbitrary knot and link topology in minimal two-band non-Hermitian systems. The central claim is that any knot or link can be encoded into a tight-binding Hamiltonian whose band degeneracy manifold in three-dimensional momentum space reproduces that topology exactly. The construction rests on Alexander's theorem—every knot or link is the closure of a braid—and proceeds through a pipeline of braid words, trigonometric braid parametrizations, braid polynomials, and semiholomorphic polynomials on $\mathcal{S}^3$. Beyond static realization, the authors demonstrate deterministic untying of knotted ELs via a single tunable parameter and propose an electro-acoustic experimental platform.

Defective Hamiltonians and exceptional lines

The framework uses a two-band Hamiltonian $$\hat{H}(\boldsymbol{k}) = \boldsymbol{d}(\boldsymbol{k})\cdot\boldsymbol{\sigma}$$ with complex $\boldsymbol{d}$-vector, where degeneracy requires both $$\mathrm{Re}[E^2] = \boldsymbol{d}_R^2 - \boldsymbol{d}_I^2 = 0$$ and $$\mathrm{Im}[E^2] = 2\boldsymbol{d}_R\cdot\boldsymbol{d}_I = 0$$. In 3D each condition defines a surface; their intersection is generically a one-dimensional exceptional line of defective (non-diagonalizable) points. The authors adopt a canonical ansatz $\boldsymbol{d}_R = [f_1, \Lambda, 0]$ and $\boldsymbol{d}_I = [0, f_2, \Lambda]$, reducing the design problem to choosing two real scalar functions $f_1$, $f_2$ whose zero-level sets intersect along the target curve. Because the degeneracy arises from the intersection of two real conditions in a generic non-Hermitian system, knotted ELs are stable without symmetry protection or fine-tuning—a structural contrast with Hermitian nodal lines, which require crystalline or chiral symmetries for their existence.

The four-step constructive pipeline

The method proceeds in four steps. First, the target knot is specified by a braid word in generators $\sigma_i^{\pm1}$. Second, a trigonometric braid is built from finite Fourier series $$\hat{H}(\boldsymbol{k}) = \boldsymbol{d}(\boldsymbol{k})\cdot\boldsymbol{\sigma}$$0, $$\hat{H}(\boldsymbol{k}) = \boldsymbol{d}(\boldsymbol{k})\cdot\boldsymbol{\sigma}$$1 reproducing the crossing sequence, with $$\hat{H}(\boldsymbol{k}) = \boldsymbol{d}(\boldsymbol{k})\cdot\boldsymbol{\sigma}$$2 fixing over/under-crossing signs. Third, the braid polynomial $$\hat{H}(\boldsymbol{k}) = \boldsymbol{d}(\boldsymbol{k})\cdot\boldsymbol{\sigma}$$3 is promoted to a semiholomorphic polynomial $$\hat{H}(\boldsymbol{k}) = \boldsymbol{d}(\boldsymbol{k})\cdot\boldsymbol{\sigma}$$4 via de Moivre substitutions, whose zero set on $$\hat{H}(\boldsymbol{k}) = \boldsymbol{d}(\boldsymbol{k})\cdot\boldsymbol{\sigma}$$5 realizes the knot. The paper states this as a theorem: for every knot or link there exists such a polynomial. Fourth, a composite mapping $$\hat{H}(\boldsymbol{k}) = \boldsymbol{d}(\boldsymbol{k})\cdot\boldsymbol{\sigma}$$6 embeds the construction into Bloch momentum space, yielding explicit tight-binding Hamiltonians. The degree of $$\hat{H}(\boldsymbol{k}) = \boldsymbol{d}(\boldsymbol{k})\cdot\boldsymbol{\sigma}$$7 bounds the hopping range—for instance, the Hopf link ($$\hat{H}(\boldsymbol{k}) = \boldsymbol{d}(\boldsymbol{k})\cdot\boldsymbol{\sigma}$$8) requires only next-nearest-neighbor couplings.

A practical advantage claimed over prior algorithmic constructions is conciseness: for short symmetric braids the resulting parametrizations have few Fourier terms and low order, e.g., only two terms each in $$\hat{H}(\boldsymbol{k}) = \boldsymbol{d}(\boldsymbol{k})\cdot\boldsymbol{\sigma}$$9 and $\boldsymbol{d}$0 for the knot $\boldsymbol{d}$1.

Realized examples

The framework is validated across a curated sequence of increasing complexity:

Topology	Braid word	Class
Trefoil $\boldsymbol{d}$2	$\boldsymbol{d}$3	Torus knot
Hopf link $\boldsymbol{d}$4	$\boldsymbol{d}$5	Torus link
Figure-eight $\boldsymbol{d}$6	$\boldsymbol{d}$7	Lemniscate knot
$\boldsymbol{d}$8	$\boldsymbol{d}$9	Lemniscate knot
Borromean link $$\mathrm{Re}[E^2] = \boldsymbol{d}_R^2 - \boldsymbol{d}_I^2 = 0$$0	$$\mathrm{Re}[E^2] = \boldsymbol{d}_R^2 - \boldsymbol{d}_I^2 = 0$$1	Lemniscate link
Three-twist $$\mathrm{Re}[E^2] = \boldsymbol{d}_R^2 - \boldsymbol{d}_I^2 = 0$$2	$$\mathrm{Re}[E^2] = \boldsymbol{d}_R^2 - \boldsymbol{d}_I^2 = 0$$3	Non-fibred knot
Miller-Institute $$\mathrm{Re}[E^2] = \boldsymbol{d}_R^2 - \boldsymbol{d}_I^2 = 0$$4	$$\mathrm{Re}[E^2] = \boldsymbol{d}_R^2 - \boldsymbol{d}_I^2 = 0$$5	Fibred hyperbolic knot
Whitehead link $$\mathrm{Re}[E^2] = \boldsymbol{d}_R^2 - \boldsymbol{d}_I^2 = 0$$6	$$\mathrm{Re}[E^2] = \boldsymbol{d}_R^2 - \boldsymbol{d}_I^2 = 0$$7	Two-component link

For torus knots the construction collapses to the holomorphic polynomial $$\mathrm{Re}[E^2] = \boldsymbol{d}_R^2 - \boldsymbol{d}_I^2 = 0$$8, and the number of link components equals $$\mathrm{Re}[E^2] = \boldsymbol{d}_R^2 - \boldsymbol{d}_I^2 = 0$$9, matching the classical classification. Notably, knots $$\mathrm{Im}[E^2] = 2\boldsymbol{d}_R\cdot\boldsymbol{d}_I = 0$$0 and $$\mathrm{Im}[E^2] = 2\boldsymbol{d}_R\cdot\boldsymbol{d}_I = 0$$1 share the same unsigned braid word but differ in crossing signs; the framework handles this by reusing $$\mathrm{Im}[E^2] = 2\boldsymbol{d}_R\cdot\boldsymbol{d}_I = 0$$2 while redesigning $$\mathrm{Im}[E^2] = 2\boldsymbol{d}_R\cdot\boldsymbol{d}_I = 0$$3, illustrating how the sign structure of braiding maps directly onto Hamiltonian parameters. In every case the equiphase surface $$\mathrm{Im}[E^2] = 2\boldsymbol{d}_R\cdot\boldsymbol{d}_I = 0$$4 yields a Seifert surface bounded by the EL, confirming the metallic character of these phases—the Seifert surface encodes the defective nature of the degeneracy, a feature absent in Hermitian nodal-line systems.

Continuous untying via parameter tuning

A second major result is that knotted ELs can be untied deterministically by increasing the scaling parameter $$\mathrm{Im}[E^2] = 2\boldsymbol{d}_R\cdot\boldsymbol{d}_I = 0$$5, which controls the effective width of the braid trajectory. For the Figure-eight knot, increasing $$\mathrm{Im}[E^2] = 2\boldsymbol{d}_R\cdot\boldsymbol{d}_I = 0$$6 from 1 through critical values $$\mathrm{Im}[E^2] = 2\boldsymbol{d}_R\cdot\boldsymbol{d}_I = 0$$7 and $$\mathrm{Im}[E^2] = 2\boldsymbol{d}_R\cdot\boldsymbol{d}_I = 0$$8 drives reconnection events mediated by the motion, merging, and splitting of exceptional points, transiently forming exceptional chains (ECs), and ending in an unknotted ring. For the Three-twist knot $$\mathrm{Im}[E^2] = 2\boldsymbol{d}_R\cdot\boldsymbol{d}_I = 0$$9, critical values near $\boldsymbol{d}_R = [f_1, \Lambda, 0]$0 and $\boldsymbol{d}_R = [f_1, \Lambda, 0]$1 successively simplify the braid word to yield three fully unlinked rings for $\boldsymbol{d}_R = [f_1, \Lambda, 0]$2. The vortices of $\boldsymbol{d}_R = [f_1, \Lambda, 0]$3 in momentum-space slices track the underlying EP trajectories throughout.

The authors are explicit that these pathways need not be minimal in the sense of the unknotting number—both $\boldsymbol{d}_R = [f_1, \Lambda, 0]$4 and $\boldsymbol{d}_R = [f_1, \Lambda, 0]$5 have unknotting number 1, so mathematically shorter paths exist—but the induced protocol is continuous, robust, and experimentally feasible, requiring no fine-tuning beyond monotonic variation of a single control knob mappable to gain/loss strength or synthetic-dimension couplings. This positions knot complexity as a tunable physical resource rather than a fixed topological label.

Proposed acoustic realization

The authors propose implementing the required nonreciprocal complex couplings via electro-acoustic feedback: paired cavities equipped with microphone-speaker links processed by amplification and phase-shifting circuits realize unequal forward and backward couplings $\boldsymbol{d}_R = [f_1, \Lambda, 0]$6. The braid-width parameter $\boldsymbol{d}_R = [f_1, \Lambda, 0]$7 would be tuned through circuit gain and phase offsets, with detection via microphone arrays or laser vibrometry. The proposal remains schematic; feedback stability and circuit noise are acknowledged as unresolved challenges, and no quantitative tolerance analysis is provided.

Limitations and open questions

Several limitations bear directly on the results. The construction is confined to two-band models; extension to multiband systems with higher-order exceptional degeneracies and band permutations is left open. The interplay of knotted ELs with crystalline, PT, and non-Hermitian chiral symmetries—which could refine the classification or protect subclasses—is unexplored. The untying protocol's dependence on the specific polynomial family means the reachable transition pathways are not exhaustive, and whether distinct unknotting routes carry universal spectral signatures or are selected by material-specific constraints remains an open question raised explicitly by the authors. Finally, dynamical responses associated with knotted ELs—adiabatic encircling, non-Hermitian pumping, nonlinear spectral response—are identified as directions rather than results.

Conclusion

This work establishes a systematic correspondence between arbitrary braid-encoded knot and link topology and non-Hermitian band degeneracies in minimal two-band lattice models, with stability guaranteed generically rather than by symmetry. The combination of constructive universality, explicit tight-binding realizations, single-parameter topological transitions, and a concrete acoustic implementation provides a coherent program for treating knot invariants—braid words, Alexander polynomials, Seifert surfaces—as physical classifiers of non-Hermitian metallic phases.