# Classification of Exceptional Nodal Topologies Protected by $\mathcal{PT}$
Symmetry

Marcus Stålhammar  Emil J. Bergholtz  Department of Physics, Stockholm
University, AlbaNova University Center, 106 91 Stockholm, Sweden

###### Abstract

Exceptional degeneracies, at which both eigenvalues and eigenvectors coalesce,
and parity-time ($\operatorname{\mathcal{PT}}$) symmetry, reflecting balanced
gain and loss in photonic systems, are paramount concepts in non-Hermitian
systems. We here complete the topological classification of exceptional nodal
degeneracies protected by $\operatorname{\mathcal{PT}}$ symmetry in up to
three dimensions and provide simple example models whose exceptional nodal
topologies include previously overlooked possibilities such as second-order
knotted surfaces of arbitrary genus, third-order knots and fourth-order
points.

##  I Introduction

Parity-time ($\operatorname{\mathcal{PT}}$) symmetric non-Hermitian (NH)
descriptions were originally suggested as a fundamental amendment to the
standard quantum mechanics motivated by their capacity of having real spectra
benderboettcher ; bender . By now, NH models are instead recognized as
effective descriptions with a remarkably wide range of applications NHbook ,
and $\operatorname{\mathcal{PT}}$ symmetry is an ubiquitous feature of
photonic experiments where it reflects a balance between gain and loss
OzRoNoYa2019 . Exceptional point (EP) degeneracies at which both eigenvalues
and eigenvectors coalesce is a central and uniquely NH feature associated with
a rich phenomenology in these systems EPreview . Recently, the merger between
topological hasankane ; qizhang  and NH physics has led to the burgeoning
field of NH topological systems review .

The topological properties of NH nodal band structures, consisting of EPs, are
central topic within this field review . EPs are topologically distinguished
from Hermitian degeneracies in that they are generic in the bulk of two-
dimensional (2D) systems BerryDeg ; Heiss ; koziifu ; NHarc , giving closed
(potentially knotted) lines of EPs in 3D EPrings ; CaBe2018 ; EPringExp ;
ourknots ; ourknots2 ; chinghuaknot ; yangknot . In general, EPs of order $n$
($n$EPs) require the tuning of $2n-2$ real parameters, resulting in their
absence for $n\geq 3$ in systems of 3D or lower. The inclusion of discrete
symmetries can, however, relax this constraint and make symmetry-protected
2EPs generic already in one dimension (1D), while rings and surfaces of 2EPs
are stabilized in 2D and 3D, respectively Budich2019 ; Yoshida2019 ; Okugawa ;
Zhou ; HCphot ; Etorus . These EPs are associated with open Fermi arc
degeneracies in the real part of the spectrum with telltale signatures in
experiments NHarc . Very recently, it was shown that physically relevant
symmetries can also stabilize higher-order EPs, and explicit examples of
models with point-like 3EPs in 2D were explored 3Tsuneya ; ips3 .

Figure 1: $\operatorname{\mathcal{PT}}$ symmetry protected exceptional nodal
degeneracies in three-dimensional (3D) systems. In (a) and (b), second-order
exceptional surfaces of the Hamiltonian in Eq. (4) are displayed. (a) An
inflated exceptional trefoil knot. (b) An inflated Hopf-link. The surfaces are
obtained by adding a symmetry-preserving perturbation to systems hosting
ordinary nodal knot degeneracies. The surfaces (a) and (b) are of genus 1 and
2, respectively, and higher genus surfaces can be obtained in a systematic
manner. (c) An exceptional trefoil knot of order three is shown, which
generically occurs in $\operatorname{\mathcal{PT}}$-symmetric models with
three or more bands, e.g., Eq. (5). Lastly, exceptional points of order four
are displayed in (d). Together, these exceptional structures provide a
complete classification of exceptional nodal topologies protected by
$\operatorname{\mathcal{PT}}$ symmetry in 3D, cf. Table 1.

Here, we show that $\operatorname{\mathcal{PT}}$ symmetry protects a
remarkably rich family of nodal EP topologies in 3D. By explicit construction,
we below provide a classification thereof, including possibilities such as
knotted surfaces of arbitrary genus of 2EPs, knotted lines of 3EPs and point-
like 4EPs, cf. Figure 1 and Table 1. All structures are generic and stable in
the sense that they are indeed protected by $\operatorname{\mathcal{PT}}$
symmetry; small perturbations can destroy the EP topologies only if they break
$\operatorname{\mathcal{PT}}$. This points to their experimental relevance
mainly in photonic systems where the $\operatorname{\mathcal{PT}}$ can be
realized with great precision and system parameters may be tuned with
excellent control OzRoNoYa2019 .

Exceptional Nodal Topologies
---
Dimension | Without $\operatorname{\mathcal{PT}}$ | With $\operatorname{\mathcal{PT}}$
$D=1$ | — | 2EPs
$D=2$ | 2EPs | Lines of 2EPs
3EPs
$D=3$ | Knots of 2EPs | Surfaces of 2EPs
Knots of 3EPs
4EPs
Table 1: The impact of $\operatorname{\mathcal{PT}}$ symmetry on the
exceptional nodal topologies in systems of physically relevant dimensions.

##  II $\operatorname{\mathcal{PT}}$ and codimension of $N$EPs

The codimension of $n$EPs is related to the properties of the corresponding
characteristic polynomial $\operatorname{\text{Ch}}(\lambda)$. Imposing
$\operatorname{\mathcal{PT}}$ symmetry yields the following constraints on
$\operatorname{\text{Ch}}(\lambda)$ 3Tsuneya :

$\displaystyle\operatorname{\text{Ch}}_{\operatorname{\mathcal{PT}}}(\lambda)$
$\displaystyle=\det\left(H_{\operatorname{\mathcal{PT}}}-\lambda\right)=\det\left(H_{\operatorname{\mathcal{PT}}}^{*}-\lambda\right),$
(1)

where ∗ denotes complex conjugation. Explicitly, Eq. (1) reads,

$\operatorname{\text{Ch}}_{\operatorname{\mathcal{PT}}}(\lambda)=\sum_{j=0}^{n}a_{j}\lambda^{j}=\sum_{j=0}^{n}a_{j}^{*}\lambda^{j}.$
(2)

Consequently, $a_{j}$ are real-valued functions, indicating that
$\operatorname{\mathcal{PT}}$ reduces the number of real constraints for
$n$EPs to $n-1$, thus requiring the tuning of at least $n-1$ real parameters.
Below, we show by explicit construction that these real constraints indeed
have non-trivial real-valued solutions in 3D, going beyond earlier works such
as Ref. 3Tsuneya . The general argument confirms that a further decrease of
the codimension of the EPs are not stabilizedby $\operatorname{\mathcal{PT}}$
and therefore relies on fine-tuning. Consequently, our findings constitute a
classification of EP topologies protected by $\operatorname{\mathcal{PT}}$
symmetry in 3D since the exceptional structures map over to embeddings of the
respective dimension, ensuring that all corresponding topologies can be
obtained algknot ; knotnodal ; Milnor .

Throughout this Letter, we use a representation such that
$\operatorname{\mathcal{PT}}$ symmetry amounts to complex conjugation, meaning
that matrix representations of $\operatorname{\mathcal{PT}}$-symmetric
Hamiltonians will consist of purely real entries and
$(\operatorname{\mathcal{PT}})^{2}=1$.

Figure 2: Second-order exceptional surfaces of the Hamiltonian in Eq. (4). In
all panels, $M=0.5$. In (a), $p=q=1$ and $\delta=0.4$, resulting in an
exceptional torus. In (b), $p=3$, $q=2$ and $\delta=0.15$, resulting in an
inflated exceptional trefoil knot. Despite appearing topologically different,
both (a) and (b) display genus 1 surfaces, since the inflated trefoil knot is
an embedding of the torus. In (c), $p=q=3$ and $\delta=0.4$, resulting in a
genus 3 surface. In this controlled manner, exceptional nodal surfaces of any
topology can be achieved by choosing $p=q=n$.

##  III Exceptional nodal surfaces

$\operatorname{\mathcal{PT}}$ symmetry preserving anti-Hermitian terms split
ordinary Hermitian nodal points to exceptional rings in two-band systems
Budich2019 . Generalizing this reasoning to higher dimensions, nodal lines
split to exceptional surfaces when exposed to similar perturbations. To
demonstrate this, we first show that $\operatorname{\mathcal{PT}}$ symmetry-
protected nodal torus knots are split into exceptional knotted tori. Consider
the following $\operatorname{\mathcal{PT}}$-symmetric Hermitian Hamiltonian
hermknot ,

$\displaystyle H_{(p,q)}$
$\displaystyle=\text{Re}\left[\zeta^{p}(k_{x},k_{y},k_{z})+\xi^{q}(k_{x},k_{y},k_{z})\right]\sigma^{x}$
$\displaystyle+\text{Im}\left[\zeta^{p}(k_{x},k_{y},k_{z})+\xi^{q}(k_{x},k_{y},k_{z})\right]\sigma^{z},$
(3)

where a $\sigma^{y}$-term is absent due to the symmetry. Here,
$\zeta(k_{x},k_{y},k_{z})=k_{x}+ik_{y}$,
$\xi=k_{z}+i\left[M-(k_{x}^{2}+k_{y}^{2}+k_{z}^{2})\right]$,
$M\in\operatorname{\mathbb{R}}$, $\sigma^{i}$ the Pauli matrices, and $k_{i}$
the lattice momentum components. Here, $p$ and $q$ are positive integers, and
the corresponding nodal points form torus knots when $p$ and $q$ are
relatively prime, and torus links otherwise. By adding a constant symmetry
preserving anti-Hermitian term, resulting in the following Hamiltonian,

$H=H_{(p,q)}+i\delta\sigma^{y},$ (4)

for $\delta\in\operatorname{\mathbb{R}}$, the nodal knots are inflated to form
second-order exceptional surfaces of knotted tori, displayed in Figs. 1(a),
2(a), and (b). This construction is however not restricted to torus knots, but
is also applicable for the hyperbolic Turk’s head knots by following the
construction in Ref. ourknots2 . Details can be found in the online
Supplemental Material SuppMat .

Despite being topologically different, in the sense that the different knotted
tori cannot be continuously deformed into each other, they are all different
examples of embeddings of the regular torus into
$\operatorname{\mathbb{R}}^{3}$. Thus, the genus of a knotted torus is alway
$1$—the genus is independent of the embedding. Since embeddings of 2D surfaces
in 3D spaces are completely classified by the corresponding genus, we expect
higher genus exceptional surfaces to appear as well. To this end, let us again
consider the Hamiltonian in Eq. (4), and let $p=q=n$. The exceptional
structure for $H$ consist of $n$ individually linked tori. When $\delta$
becomes large enough, the tori are merged together close to the origin of
momentum space, and the exceptional structure attain the form of a sphere with
$n$ handles, i.e., a genus $n$ surface. This is explicitly shown in Figs. 1
(b) and 2 (c) for $n=2$ and $n=3$, respectively, while higher genus examples
are provided in the Supplemental Material SuppMat . Thus, it is possible to
obtain second-order exceptional surfaces with any genus. These surfaces are
furthermore generic and stable, in the sense that they are protected by
$\operatorname{\mathcal{PT}}$ symmetry.

##  IV Knotted exceptional lines of order three

The plethora of exceptional nodal topologies protected by
$\operatorname{\mathcal{PT}}$ symmetry does not only contain second-order
surfaces, but also higher-order degeneracies. Below, building on the recent
examples on 3EPs (points) in two dimensions ips3 ; 3Tsuneya , we exemplify
this by explicit construction of a three-band model hosting knotted
exceptional lines of order three. Consider the Hamiltonian given by

$H_{3}=\begin{pmatrix}f_{2}&0&\Lambda\\\ \alpha&0&\beta\\\
f_{2}&f_{1}&-f_{2}\end{pmatrix},$ (5)

where $f_{1}$ and $f_{2}$ are continuously differentiable functions of the
lattice momentum components, and
$\alpha,\beta,\Lambda\in\operatorname{\mathbb{R}}$. The corresponding
characteristic equation determining the eigenvalues reads,

$\displaystyle\quad f_{1}\left(\alpha\,\Lambda-\beta\,f_{2}\right)+$
$\displaystyle+\lambda\left(f_{2}^{2}+\Lambda\,f_{2}+\beta\,f_{1}\right)-\lambda^{3}=0.$
(6)

To have third-order exceptional structures, all eigenvalues have to coalesce.
This happens exactly when

$\displaystyle f_{1}\left(\alpha\,\Lambda-\beta\,f_{2}\right)=0,$ (7)
$\displaystyle\left(f_{2}^{2}+\Lambda\,f_{2}+\beta\,f_{1}\right)=0.$ (8)

When $|\Lambda|$ is sufficiently large, the only solution becomes
$f_{1}=f_{2}=0$. Recalling the interpretation of a knot as the intersection of
two implicitly defined surfaces, $f_{1}$ and $f_{2}$ can be defined such that
their common zeros resemble any knot algknot ; knotnodal . By choosing

$\displaystyle f_{1}$
$\displaystyle=\text{Re}\left[\zeta^{p}(k_{x},k_{y},k_{z})+\xi^{q}(k_{x},k_{y},k_{z})\right],$
(9) $\displaystyle f_{2}$
$\displaystyle=\text{Im}\left[\zeta^{p}(k_{x},k_{y},k_{z})+\xi^{q}(k_{x},k_{y},k_{z})\right],$
(10)

with $\zeta$ and $\xi$ as above, the exceptional lines are torus knots Milnor
, while an exceptional hyperbolic figure-eight knot is obtained when

$\displaystyle f_{1}$
$\displaystyle=(k_{y}^{2}-k_{z}^{2})\epsilon^{2}+R\left(8R^{2}-2\epsilon^{2}\right),$
(11) $\displaystyle f_{2}$
$\displaystyle=2\sqrt{2}R\,k_{y}\,k_{z}+k_{x}\left(8R^{2}-2\epsilon^{2}\right),$
(12)

where $R^{2}=\epsilon^{2}-\left(k_{x}^{2}+k_{y}^{2}+k_{z}^{2}\right)$
frenchknot . In Fig. 3, a sample of illustrations is provided, including
examples from both of the cases above. The topology of the knots can be
determined by computing knot invariant polynomials, such as the Alexander
polynomial ourknots2 ; CH4D . In the Supplemental Material SuppMat ,
additional examples, including hyperbolic Turk’s head knots, are provided.

Figure 3: Knotted and linked exceptional lines of order three appearing as
triple eigenvalue degeneracies in the model described by Eq. (5). For all
panels, $\alpha=1$, $\beta=-1$ and $\Lambda=100$. (a) An exceptional Hopf-
link, which corresponds to $p=q=2$ in Eqs. (9) and (10). (b) An exceptional
trefoil knot, for which $p=3$ and $q=2$. In both cases, $M=0.5$. The
exceptional knot in (c) is a figure-eight knot, appearing when choosing
$f_{1}$ and $f_{2}$ as in Eqs. (11) and (12), with $\epsilon=1$.

To emphasize that the knotted exceptional structures are indeed of order
three, let us consider the eigenvectors, which are given by

$V_{i}=\left(-\frac{\Lambda}{f_{2}-E_{i}},-\frac{-\beta\,f_{2}+\alpha\,\Lambda+\beta\,E_{i}}{\left(f_{2}-E_{i}\right)},1\right)^{\text{T}},$
(13)

where $E_{i}$ denotes the corresponding eigenvalue, and $i=1,2,3$. Taking the
difference $V_{i}-V_{j}$ for $i\neq j$, and letting $E_{i}\to E_{j}$, the
eigenvectors indeed coalesce when the eigenvalues do, and hence, the
degeneracies at $E_{i}=0$ are exceptional of order three. They are furthermore
stable towards symmetry-preserving perturbations, which follows by the
stability of the models in Refs. ourknots ; ourknots2 .

##  V 4EPs in perturbed Dirac semimetals

Figure 4: 4EPs appearing in the system described by the Hamiltonian in Eq.
(15). In both panels, $\delta_{4}=0.8$‚ $\delta_{6}=1$ and $\delta_{12}=1.1$.
In (a), $k_{x}=0$, and the 4EPs occur at
$(k_{y},k_{z})=(\pm_{1}\delta_{6},\pm_{2}\delta_{4})$, where $\pm_{1}$ and
$\pm_{2}$ are independent of each other. In (b), $k_{z}=0$ and the 4EPs occur
at $(k_{x},k_{y})=(\pm_{1}0.669048,\pm_{2}1.31681)$. Since 4EPs necessarily
occur at $E=0$ they correspond exactly to when the maximum of the absolute
value of all eigenvalues vanish simultaneously. These points are furthermore
stable towards any small, but finite, perturbation, and are thus generic in
systems protected by $\operatorname{\mathcal{PT}}$ symmetry.

Finally, we demonstrate the existence of 4EPs in $\mathcal{PT}$-symmetric
four-band systems. Let us start with the Hamiltonian of a real Dirac
semimetal,

$H_{\operatorname{\text{D}}}=k_{x}\Gamma^{1}+k_{y}\Gamma^{2}+k_{z}\Gamma^{3},$
(14)

where $\Gamma^{1}=\sigma_{1}\otimes\tau_{0}$,
$\Gamma^{2}=\sigma_{2}\otimes\tau_{2}$ and
$\Gamma^{3}=\sigma_{3}\otimes\tau_{0}$, together with
$\Gamma^{4}=\sigma_{2}\otimes\tau_{1}$ and
$\Gamma^{5}=\sigma_{2}\otimes\tau_{3}$ constitute a representation of the
Clifford algebra. Here, $\sigma_{i}$ and $\tau_{i}$ denote the Pauli matrices.
Anti-commuting $\Gamma^{1},...,\Gamma^{4}$ provides a complete basis of
$4\times 4$-matrices. Adding an anti-Hermitian term on the form
$H_{\text{AH}}=i\delta_{4}\Gamma^{4}+\delta_{6}\Gamma^{6}+k_{x}\delta_{12}\Gamma^{12}$,
to obtain

$H=H_{\operatorname{\text{D}}}+H_{\text{AH}},$ (15)

with $\Gamma^{6}=\Gamma^{[1}\Gamma^{2]}$,
$\Gamma^{12}=\Gamma^{[1}\Gamma^{2}\Gamma^{3]}$ and
$\delta_{4},\delta_{6},\delta_{12}\in\operatorname{\mathbb{R}}$, splits the
Dirac node at energy $E=0$ into eight 4EPs, all located at $E=0$. Here,
$A^{[\mu_{1}}\cdot...\cdot
A^{\mu_{n}]}=\frac{1}{n!}\epsilon_{\mu_{1}...\mu_{n}}A^{\mu_{1}}\cdot...\cdot
A^{\mu_{n}}$ with $\epsilon_{\mu_{1}...\mu_{n}}$ the Levi-Civita symbol. This
can be seen by studying the characteristic polynomial, which attains the form
of a depressed quartic. If such polynomials have four-fold degenerate
solutions, they are necessarily located at $E=0$. In Fig. 4, the maximum of
the absolute values of all eigenvalues $E_{1},...,E_{4}$ are displayed as a
density plot in the appropriate planes in momentum space. The 4EPs occur
exactly when the maximum value is zero since then all eigenvalues necessarily
coalesce at $E=0$. The points are indeed exceptional since the dimensions of
the eigenspaces at those points are one, indicating that the eigenvalues
$E_{1}=E_{2}=E_{3}=E_{4}=0$ share one eigenvector.

##  VI Discussion

In this Letter, we provide a classification of exceptional nodal topologies
protected by $\operatorname{\mathcal{PT}}$ symmetry in 3D. Through explicit
construction, we demonstrate the existence of several hitherto overlooked
types of exceptional nodal degeneracies: knotted surfaces of arbitrary genus
of 2EPs, knotted lines of 3EPs, and 4EPs, thus extending the already rich
plethora of NH topological phases review ; Budich2019 ; Yoshida2019 ; Okugawa
; Zhou ; gong ; lieu2 ; Kawabata2019 ; ZhLe2019 ; KawabataExceptional ;
esakisatohasebekohmoto ; lee ; BBC ; yaowang . We note that higher-order EPs
require tuning of more parameters than allowed by dimensional constraints,
making the list of exceptional nodal topologies complete for physically
relevant systems if additional tuning variables are not considered.

We emphasize again that our findings rely only on the presence of
$\operatorname{\mathcal{PT}}$ symmetry. Being arguably the most widely
implemented symmetry in photonic experimental setups, make them highly
relevant in optics OzRoNoYa2019 . It should be possible to experimentally
simulate the models we present using single-photon interferometry as in Ref.
knotexp . We furthermore note that 2EPs, together with their bulk Fermi arcs,
as well as the more exotic knotted lines of 2EPs and their concomitant Fermi-
Seifert surfaces were both experimentally realized and observed shortly after
being theoretically predicted NHarc ; knotexp . This suggests that topological
phases of higher-order EPs are within experimental reach in several platforms
and encourages the search for tailor-made setups where they can be realized.

_Acknowledgments.—_ We are grateful to Lukas Rødland and Gregory Arone for
fruitful discussions. We also thank Jan Budich, Ipsita Mandal and Johan
Carlström for related collaborations. The authors are supported by the Swedish
Research Council (VR) and the Wallenberg Academy Fellows program as well as
the project Dynamic Quantum Matter of the Knut and Alice Wallenberg
Foundation.

_Note added:_ An independent work shows how symmetry-protected 4EPs may
naturally be realized in three-dimensional correlated many-body systems note .

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##  Appendix A Supplementary Material for “Classification of Exceptional Nodal
Topologies Protected by $\mathcal{PT}$ Symmetry”

In this supplementary material, we provide details on our derivations and
arguments presented in the main text, along with complementary figures.

###  A.1 Hyperbolic nodal knots

Here, we provide a short summary on how to construct functions whose common
zeros represent hyperbolic Turk’s head knots. For further details, see Ref.
ourknots2, .

In contrast to torus knots, which are described as zeros of a complex
polynomial Milnor , the Turk’s head knots are illustrated as the zeros of a
real polynomial, constrained on $S^{3}_{\epsilon}$ (the three-sphere of radius
$\epsilon$) frenchknot . Explicitly, the polynomial is,

$F\left\\{x,y,\text{Re}\left[(z+it)^{N}\right],\text{Im}\left[(z+it)^{N}\right]\right\\},$
(16)

with

$\displaystyle F(x,y,z,t)$
$\displaystyle=\left\\{z\left[x^{2}+y^{2}+z^{2}+t^{2}\right]+x\left[8x^{2}-2\left(x^{2}+y^{2}+z^{2}+t^{2}\right)\right],\right.$
$\displaystyle\quad\left.\sqrt{2}t\,x+y\left[8x^{2}-\left(x^{2}+y^{2}+z^{2}+t^{2}\right)\right]\right\\},$
(17)

where $x,y,z,t$ denote coordinates on $\operatorname{\mathbb{R}}^{4}$ and
$N\in\mathbb{N}$. Note that
$F:\operatorname{\mathbb{R}}^{4}\to\operatorname{\mathbb{R}}^{2}$. By finding
the zeros, restricting them to $S^{3}_{\epsilon}$, and introducing the lattice
momentum appropriately, these knots can be realized as third-order exceptional
degeneracies. In Figure 7, this is illustrated for different values of $N$,
with the lattice momentum introduced as,

$x=\epsilon^{2}-\left(k_{x}^{2}+k_{y}^{2}+k_{z}^{2}\right),\quad y=k_{x},\quad
z=k_{y},\quad t=k_{z}.$ (18)

Additionally, these hyperbolic knots can be inflated to form knotted surfaces,
just as displayed for the torus knots in the main text. This is illustrated in
Figure 5, along with a more complicated inflated torus knot.

###  A.2 Higher genus surfaces

Figure 5: Exceptional knotted surfaces of order 2. (a) displays an inflated
$(4,3)$ torus knot, obtained from the Hamiltonian in Eq. (4), with $p=4$,
$q=3$, $M=1$ and $\delta=0.35$. In (b) and (c), an inflated figure-eight knot
and inflated Borromean rings are shown, obtained by using Eqs. (16) and (A.1)
in the Hamiltonian Eq. (4), with $\delta=0.15$, $\epsilon=1$ and $N=2,3$
respectively. Figure 6: higher genus exceptional surfaces of order 2 from the
Hamiltonian Eq. (4). In all panels, $M=1$. In (a), $p=q=4$, and $\delta=1$,
giving a genus 4 surface. In (b), $p=q=6$, and $\delta=2.5$ resulting in a
genus 6 surface. Lastly, (c) displays a genus 10 surface, obtained by choosing
$p=q=10$ and $\delta=30$.

As mentioned in the main text, the presented techniques can in principle be
used to achieve second-order exceptional surfaces of arbitrary genus. Here, we
present a few additional figures to strengthen this claim. This is done by
using the Hamiltonian on the form,

$H_{2}=\text{Re}\left[\zeta^{p}(k_{x},k_{y},k_{z})+\xi^{q}(k_{x},k_{y},k_{z})\right]\sigma^{x}+\text{Im}\left[\zeta^{p}(k_{x},k_{y},k_{z})+\xi^{q}(k_{x},k_{y},k_{z})\right]\sigma^{z}+i\delta\sigma^{y},$
(19)

taking $p=q=n$, and $\delta\in\operatorname{\mathbb{R}}$. In Figure 6, we
illustrate this for $n=4,6,10$, giving exceptional surfaces of genus 4, 6 and
10 respectively.

###  A.3 Controlling topology of third-order exceptional lines

Figure 7: Hyperbolic exceptional knots of order 3 of the Hamiltonian Eq. (5),
when defining $f_{1}$ and $f_{2}$ in terms of the components of the polynomial
$F$, defined in Eqs. (16) and (A.1). For all panels, $\alpha=1$, $\beta=-1$
and $\Lambda=100$, while $N=3,4,5$ in (a), (b) and (c) respectively. (a)
displays Borromean rings, while (b) and (c) are mote complicated hyperbolic
knots in the family of Turk’s head knots.

Consider an arbitrary real $3\times 3$ Hamiltonian,

$H_{3}=\begin{pmatrix}a_{1}&a_{2}&a_{3}\\\ a_{4}&a_{5}&a_{6}\\\
a_{7}&a_{8}&a_{9}\end{pmatrix}$ (20)

with all $a_{j}$ being real-valued, continuously differentiable function of
the lattice momentum components, $j=1,...,9$. Since we are interested in when
all the eigenvalues of $H_{3}$ coalesce, we can assume that $H_{3}$ is
traceless and take $a_{5}=0$, $a_{1}=-a_{9}$. The corresponding characteristic
polynomial reads,

$\operatorname{\text{Ch}}_{H_{3}}(\lambda)=a_{2}a_{6}a_{7}+a_{3}a_{4}a_{8}-a_{1}a_{6}a_{8}+a_{1}a_{2}a_{4}+\lambda\left(a_{2}a_{4}+a_{3}a_{7}+a_{6}a_{8}-a_{1}^{2}\right)-\lambda^{3}$
(21)

Triple eigenvalue degeneracies occur when,

$\displaystyle
a_{2}a_{6}a_{7}+a_{3}a_{4}a_{8}-a_{1}a_{6}a_{8}+a_{1}a_{2}a_{4}$
$\displaystyle=0,$ (22) $\displaystyle
a_{2}a_{4}+a_{3}a_{7}+a_{6}a_{8}-a_{1}^{2}$ $\displaystyle=0.$ (23)

Taking $a_{2}=0$ gives us

$\displaystyle a_{8}\left(a_{3}a_{4}-a_{1}a_{6}\right)$ $\displaystyle=0,$
(24) $\displaystyle a_{3}a_{7}+a_{6}a_{8}-a_{1}^{2}$ $\displaystyle=0.$ (25)

We note that Eq. (24) is satisfied if either $a_{8}=0$ or
$a_{3}a_{4}-a_{1}a_{6}=0$. Inspired by Ref. ourknots, , we let $|a_{3}|\gg 0$,
and thus forcing $a_{8}=0$ in order to Eq. (24) to hold. By furthermore
letting $a_{1}=a_{7}$, Eqs. (24) and (25) hold when $a_{1}=a_{8}=0$. Recalling
the constructions in Refs. ourknots, ; ourknots2, , we note that by letting
the common zeros of $a_{1}$ and $a_{8}$ represent knots, the resulting
exceptional line will indeed be knotted. Let $f_{1}$ and $f_{2}$ denote such
functions, define $a_{1}:=f_{2}$, $a_{8}=f_{1}$, and let $a_{3}=\Lambda$,
$a_{4}=\alpha$ and $a_{6}=\beta$, with
$\alpha,\beta,\Lambda\in\operatorname{\mathbb{R}}$, and $|\Lambda|\gg 0$.
Then, the resulting Hamiltonian,

$H_{3}=\begin{pmatrix}f_{2}&0&\Lambda\\\ \alpha&0&\beta\\\
f_{2}&f_{1}&-f_{2}\end{pmatrix}$ (26)

host third-order exceptional knots. Torus knots are obtained when, e.g.,
$f_{1}=\text{Re}\left[\zeta^{p}(k_{x},k_{y},k_{z})+\xi^{q}(k_{x},k_{y},k_{z})\right]$
and
$f_{2}=\text{Im}\left[\zeta^{p}(k_{x},k_{y},k_{z})+\xi^{q}(k_{x},k_{y},k_{z})\right]$,
as discussed in the main text, while hyperbolic Turk’s head knots are obtained
by choosing $f_{1}$ and $f_{2}$ as the first and second components of the
function $F$ defined in Eqs. (16) and (A.1).