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.

Figures (11)

• Figure 1: The exceptional loops (ELs) with torus knots/links $T(p, q)$ configurations (main panels) and their corresponding Seifert surfaces (insets) in tight-binding models. The Seifert surfaces, whose boundaries are knotted and linked ELs, is obtain from the equiphase surface $arg(f_a) = \pi$. The fundamental distinction between the ELs with torus knots/links configurations is determined by the greatest common divisor $r = gcd(p,q)$. (a-d) Torus knots if $r=1$. (e-h) Torus links if $r>1$. The values of $p$ and $q$ are given at the lower right corner of each panel. $(P,Q)=(1,2.5)$ is adopted for all panels except panels (e) and (j) with $(P, Q)=(2,5)$.
• Figure 2: Constructing exceptional loops (ELs) with lemniscate knots/links configurations in tight-binding model, such as the knotted Figure-eight EL [$T(3, 2, 2)$ in the first row], the knotted $6_3$ EL [$T(5, 2, 2)$ in the second row] and linked Borromean ELs [$T(3, 3, 2)$ in the last row]. (a1) Schematic diagram of the Figure-eight knot (left) and its braid word $(\sigma_1\sigma_2^{-1})^2$ (right), which encodes the sequence of crossings for a $3$-strand braid. The Figure-eight knot $4_1$ is the the simplest non-torus knot. Strands are labeled increasing order at $t = 0$: $1$(green), $2$(red), $3$(blue). (a2) Schematic diagram of the $6_3$ knot (left) and its braid word $(\sigma_1\sigma_3^{-1}\sigma_2\sigma_4^{-1})^2$ (right), which encodes the sequence of crossings for a $5$-strand braid. Strands are labeled increasing order at $t = 0$: $1$(green), $2$(red), $3$(blue), $4$(orange), $5$(magenata). (a3) Schematic diagram of the Borromean link (left) and its braid word $(\sigma_1\sigma_2^{-1})^3$ (right), which encodes the sequence of crossings for a $3$-strand braid. Strands are labeled increasing order at $t = 0$: $1$(green), $2$(red), $3$(blue). (b1-b3) Plot of $X(nt+2\pi j)$ and $Y(nt+2\pi j)$ with $j=0,1,...,m-1$ from Eq. (\ref{['XY_lemniscate']}), giving the same unsigned braiding diagram as panels (a1-a3) for $t\in[0, 2\pi]$. The signs "$+$($-$)" in the crossings indicate over(under)-crossing of the adjacent two strands. (c1-c3) Trigonometric braid of the knots/links (solid lines) and their projection on $XY$-plane (dotted lines) obtained from Eqs. (\ref{['B_lemniscate']}-\ref{['XY_lemniscate']}). (d1-d3) The knotted $4_1$, knotted $6_3$ and linked Borromean ELs obtain from Eqs. (\ref{['Fa_K41']}) with $(a, P, Q)=(1,2,5)$, Eq. (\ref{['Fa_K63']}) with $(a,P,Q)=(1,2,4.6)$ and Eq. (\ref{['Fa_Borromean']}) with $(a, P, Q)=(1,2,4)$, respectively. Inset: The Seifert surface, whose boundaries coincide with the ELs, are obtained from the equiphase surfaces $arg(f_a)=\pi$.
• Figure 3: Constructing the knotted $5_2$ exceptional loop (EL) in tight-binding model. (a) Schematic diagram of the knot $5_2$ (left) and its braid word $\sigma_1^{-1}\sigma_2\sigma_1^{3}\sigma_2$ (right), which encodes the sequence of crossings for a $3$-strand braid. The Three-twist knot $5_2$ is the simplest non-fibred knot. Strands are labeled increasing order at $t=0$: 1(green), 2(red), 3(blue). (b) Plot of $X(t+2\pi j)$ and $Y(t+2\pi j)$ with $j=1,2,3$ from Eq. (\ref{['XY_K52']}), giving the same unsigned braiding diagram as panel (a) for $t\in[0, 2\pi]$. The signs "$+$($-$)" in the crossings indicate over(under)-crossing of the adjacent two strands. (c) Trigonometric braid of the knot $5_2$ (solid lines) and its projection on $XY$-plane (dotted lines) obtained from Eqs. (\ref{['B_K52']}-\ref{['XY_K52']}). (d) The knotted $5_2$ EL obtained from Eq. (\ref{['Fa_K52']}) with $(a,P,Q)=(1/4,1,2.1)$. Inset: The Seifert surface, whose boundary coincides with the knotted $5_2$ EL, is obtained from the equiphase surface $arg(f_a) = \pi$.
• Figure 4: Constructing the knotted $6_2$ exceptional loop (EL) in tight-binding model. (a) Schematic diagram of the knot $6_2$ (left) and its braid word $\sigma_1\sigma_2^{-1}\sigma_1^{3}\sigma_2^{-1}$ (right), which encodes the sequence of crossings for a $3$-strand braid. The Miller-Institute knot $6_2$ is a fibred and hyperbolic knot. Strands are labeled increasing order at $t=0$: 1(green), 2(red), 3(blue). (b) Plot of $X(t+2\pi j)$ and $Y(t+2\pi j)$ with $j=1,2,3$ from Eq. (\ref{['XY_K62']}), giving the same unsigned braiding diagram as panel (a) for $t\in[0, 2\pi]$. The signs "$+$($-$)" in the crossings indicate over(under)-crossing of the adjacent two strands. (c) Trigonometric braid of the knot $6_2$ (solid lines) and its projection on $XY$-plane (dotted lines) obtained from Eqs. (\ref{['B_K62']}-\ref{['XY_K62']}). (d) The knotted $6_2$ EL obtained from Eq. (\ref{['Fa_K62']}) with $(a,P,Q)=(1/4,1,2.1)$. Inset: The Seifert surface, whose boundary coincides with the knotted $6_2$ EL, is obtained from the equiphase surface $arg(f_a) = \pi$.
• Figure 5: Constructing the linked Whitehead exceptional loops (ELs) in tight-binding model. (a) Schematic diagram of the Whitehead link (left) and its braid word $\sigma_1\sigma_2^{-1}\sigma_1\sigma_2^{-1}\sigma_1$ (right), which encodes the sequence of crossings for a $3$-strand braid. Strands are labeled increasing order at $t=0$: 1(green), 2(red), 3(blue). (b) Plot of $X_1(t)$, $X_1(t+2\pi)$, $X_2(t)$ and $Y_1(t)$, $Y_1(t+2\pi)$, $Y_2(t)$ from Eq. (\ref{['XY_Whitehead']}), giving the same unsigned braiding diagram as panel (a) for $t\in[0, 2\pi]$. The signs "$+$($-$)" in the crossings indicate over(under)-crossing of the adjacent two strands. (c) Trigonometric braid of the Whitehead link (solid lines) and its projection on $XY$-plane (dotted lines) obtained from Eqs. (\ref{['B_Whitehead']}-\ref{['XY_Whitehead']}). (d) The linked Whitehead ELs obtained from Eq. (\ref{['Fa_Whitehead']}) with $(a,P,Q)=(1/4,2,4)$. Inset: The Seifert surface, whose boundary coincides with the linked Whitehead ELs, is obtained from the equiphase surface $arg(f_a) = \pi$.