# Braid Protected Topological Band Structures with Unpaired Exceptional Points

J. Lukas K. König1, Kang Yang1,2, Jan Carl Budich3, and Emil J. Bergholtz1
1Department of Physics, Stockholm University, AlbaNova University Center,
10691 Stockholm, Sweden
2 Dahlem Center for Complex Quantum Systems and Institut für Theoretische
Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany
3Institute of Theoretical Physics, Technische Universität Dresden and
Würzburg-Dresden Cluster of Excellence ct.qmat, 01062 Dresden, Germany

###### Abstract

We demonstrate the existence of topologically stable unpaired exceptional
points (EPs), and construct simple non-Hermitian (NH) tight-binding models
exemplifying such remarkable nodal phases. While fermion doubling, i.e. the
necessity of compensating the topological charge of a stable nodal point by an
anti-dote, rules out a direct counterpart of our findings in the realm of
Hermitian semimetals, here we derive how noncommuting braids of complex energy
levels may stabilize unpaired EPs. Drawing on this insight, we reveal the
occurrence of a single, unpaired EP, manifested as a non-Abelian monopole in
the Brillouin zone of a minimal three-band model. This third-order degeneracy
represents a sweet spot within a larger topological phase that cannot be fully
gapped by any local perturbation. Instead, it may only split into simpler
(second-order) degeneracies that can only gap out by pairwise annihilation
after having moved around inequivalent large circles of the Brillouin zone.
Our results imply the incompleteness of a topological classification based on
winding numbers, due to non-Abelian representations of the braid group
intertwining three or more complex energy levels, and provide insights into
the topological robustness of non-Hermitian systems and their non-Abelian
phase transitions.

Following the experimental discovery of Dirac and Weyl semimetals in solid
state materials [1, 2, 3, 4, 5], Bloch bands with topologically stable nodal
points have become a major focus of research far beyond the field of condensed
matter physics [6, 1, 2, 7, 8, 9, 5, 10, 3, 4, 11, 12]. In crystalline
systems, Bloch’s theorem requires the periodicity of the band structure in
reciprocal space. As a consequence, nodal points carrying a topological charge
must be compensated for to allow the eigenstates to seamlessly fit together at
the zone boundaries of the first Brillouin zone (BZ). A prominent example
along these lines is provided by stable Weyl nodes [13, 14, 6], which are
required to occur in pairs with opposite chirality. Under the name fermion
doubling, such constraints have been discussed for decades [15, 16].

In dissipative systems described by effective non-Hermitian (NH) operators,
exceptional points (EPs) [17, 18, 19, 20, 21, 22] represent the generic
counterpart of diagonalizable degeneracies. Since stable EPs may carry
topological charge in the sense of a relative winding in the complex energy
plane, analogous constraints to the aforementioned fermion doubling have been
reported based on $\mathbb{Z}$ discriminants [23]. However, as an additional
twist, EPs are known to be of intrinsic non-Abelian nature, as characterized
by the braid group [24, 25, 26, 27, 28, 29]. This entails unique NH
topological semimetal phases that elude elementary winding number descriptions
[30, 31, 32, 33, 34]. The fusion of nodal points, a phase transition, is
topologically path-dependent [35].

Figure 1: Illustration of key results. (a) The braiding of complex energy
levels as a function of lattice momentum is at the heart of our analysis. (b)
In the two-dimensional Brillouin zone, NH phases are characterized by two
braid elements $(a,b)$ along the noncontractible loops. For gapped systems,
$a$ and $b$ commute (_group_ commutator $aba^{-1}b^{-1}=[a,b]=\mathds{1}$).
(c) Minimal model of a band structure with a nontrivial such commutator,
hosting two unpaired EP2s. (d) While the unpaired EP2s may merge into a single
EP3, the nontrivial braid topology implies that the system remains gapless.
EPs are denoted by red dots and base points of loops by $\ast$ in all panels.
Figure 2:  Composition rule for braids. (a) Schematic: Braids around
degeneracies must add up to the boundary braid. (b) Boundary braid: Spectrum
of Eq. (5) shown along the BZ boundary, counterclockwise from $(-\pi,-\pi)$.
Associating $\sigma_{i}$ to a counterclockwise exchange of eigenvalues
$E_{i},E_{i+1}$ in the complex plane allows us to read off the boundary braid
as
$\sigma_{1}^{\vphantom{}}\sigma_{2}^{\vphantom{}}\sigma_{1}^{-1}\sigma_{2}^{-1}$.
(c) Spectrum of Eq. (5) shown on the entire BZ. The orange path corresponds to
the path traversed in (a) and (b) that gives rise to the boundary braid. The
two degeneracies marked in red carry charges $\sigma_{2}^{-1}$ and
$\sigma_{1}$, which add up to the boundary charge [see Eq. (4)]. (d) Tuning
this model to fuse the two EPs [see Eq. (6)] cannot annihilate them if the
perturbation conserves the boundary charge, which is why we call these
degeneracies unpaired. Instead a single nontrivial EP3 remains as a non-
Abelian monopole that compensates the entire braid charge of the boundary.

Here, challenging the intuitive notion of fermion doubling, we reveal the
possibility of topologically stable gapless phases in periodic systems that
can host _unpaired_ EPs, i.e., nodal points that are not gapped out when
fusing. We showcase this in a tight-binding model hosting only two EPs in its
BZ whose topological charges do not cancel. Instead of annihilating, these
unpaired EP2s fuse to a single nontrivial threefold degeneracy (EP3). We find
that these NH topological band structures are enabled by noncommuting braids
of the complex energy levels along noncontractible loops in the BZ, which
remove the need for the EPs’ topological charges to pair up and cancel (see
Fig. 1 for an illustration). The real part of the spectral gap closes along
so-called Fermi arcs, which conventionally form open lines in the BZ that
terminate in degeneracies. In the phases outlined here they can instead form
closed and even noncontractible loops in the BZ. The only way to adiabatically
gap out the nodal points is to move them along nontrivial loops in the BZ
before annihilation (see Fig. 3). Transitions between these NH phases are thus
topological, and correspond to braiding of nodal points in the BZ.

_Admissible EPs by the braid group._ In the vicinity of EPs, the eigenvalues
are generally not single valued. This is manifest in the braiding of the
complex energies when encircling an EP, and endows these degeneracies with a
topological charge [25, 26]. An illustrative example is an EP2 with dispersion
$E=\pm\sqrt{k_{x}+ik_{y}}$, where two eigenvectors and eigenvalues coalesce.
The two energy levels are swapped counterclockwise along a counterclockwise
path around the EP. General $N$-fold degeneracies carry braid invariants with
$N$ strands. The corresponding braid group $B_{N}$ is generated by elementary
braids $\sigma_{j}$ ($1\leq j\leq N-1$) that swap $E_{j}$ and $E_{j+1}$
counterclockwise in the complex plane. They satisfy the relations [36]

$\sigma_{j+1}\sigma_{j}\sigma_{j+1}=\sigma_{j}\sigma_{j+1}\sigma_{j},\
\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i}\,(|i-j|>1).$ (1)

In contrast to standard Hermitian degeneracies such as Weyl points [37], EPs
are associated with non-Abelian groups, similar to topological defects in
biaxial nematics [35, 38] or multigap phases [39, 40, 41, 42, 43]. This comes
from the non-Abelian fundamental group $\pi_{1}$ of the space
$\mathcal{W}_{N}$ formed by gapped non-Hermitian Hamiltonians [25, 26, 44].
The combination or split of EPs can be described through the standard group
action [35]. As it is non-Abelian, one needs to be careful in fixing a base
point $\ast$ when doing the group product. We call an EP trivial when a small
loop around it corresponds to a trivial braid of eigenvalues, and nontrivial
otherwise. This terminology is motivated by the fact that only trivial EPs can
be gapped out by small perturbations.

We denote the homotopy class of each loop $\gamma_{j}$ around the EP $j$ with
a common base point $\ast$ as $[\gamma_{j}]\in B_{N}$. According to the
topology of a torus, the total EPs in the BZ satisfy a non-Abelian sum rule
[45, 46] [illustrated in Fig. 2(a)],

$[\gamma_{1}]\cdot[\gamma_{2}]\cdots[\gamma_{n}]=aba^{-1}b^{-1}\equiv[a,b],$
(2)

where $[a,b]$ is the group commutator of two elements $a,b$ in $B_{N}$. They
represent how the energy is braided along the meridian and the longitude of
the torus (see Fig. 1). As a result of this sum rule, models with a nontrivial
commutator $[a,b]$ must contain degeneracies; they cannot be gapped.

As the group $B_{N}$ is non-Abelian and not free, the sum rule can sometimes
be hard to apply in practice. We can extract an easier necessary condition for
EPs by taking the Abelianization $\pi_{1}/[\pi_{1},\pi_{1}]$, which is the
first homology group $\mathbf{H}_{1}(\mathcal{W}_{N})$ [47]. Since all
generators $\sigma_{j}$ are conjugate to each other, the Abelianization of the
braid group is $\mathbb{Z}$ [36]. The Abelianized element $[\gamma]_{A}$ can
be intuitively understood as the number of band permutations along a braid
(the total number of $\sigma_{j}$’s in a braid). An $n$th root EP [48, 49, 50]
has $[\gamma]_{A}=n-1$. The right-hand side of Eq. (2) is the zero element $0$
in the homology group. The Abelianized sum rule is

$\sum_{j}[\gamma_{j}]_{A}=0,$ (3)

where the sum represents the usual group product operation for Abelian groups.
Equation (3) asserts that we cannot have a single EP with $n$th root
dispersion in a BZ. Its Abelianized charge must be compensated, e.g., by a
conjugate EP that has $[\gamma]_{A}=-(n-1)$.

However, if the net number of elementary braids around a set of EPs is zero,
they satisfy the Abelianized sum rule and can occur in a BZ. Depending on
whether $a$ and $b$ commute, their topological charges can nevertheless be
nontrivial and need not cancel out. This is a significant difference from
Abelian degeneracies, where periodic boundary conditions necessitate a trivial
bulk. In the following parts, we explicitly show the existence of nontrivial
unpaired EPs. In the Supplemental Material we further present tight-binding
models that feature topologically trivial single EPs of arbitrary order in
their BZ [44]. They can be created locally and under perturbation either split
into nontrivial paired EPs or get gapped out.

_Unpaired EPs._ EPs without partners compensating their braid charge can only
exist for models with three or more bands, since the group $B_{2}$ for two-
band systems is Abelian, rendering eqs. (2) and (3) identical. In the case of
three bands, all braids are generated by $\sigma_{1},\sigma_{2}$ and the
relation (1). Those that can be written as a commutator contain equal numbers
of crossings $\sigma_{i}$ and inverse crossings $\sigma_{j}^{-1}$. A simple
nontrivial such braid is $B=\sigma^{-1}_{2}\sigma_{1}$. By relation (1), it
can be written as a commutator,

$\displaystyle[\sigma_{1},\sigma_{2}]$
$\displaystyle=\sigma_{1}\sigma_{2}\sigma_{1}^{-1}\sigma_{2}^{-1}=\sigma_{2}^{-1}(\sigma_{2}\sigma_{1}\sigma_{2})\sigma_{1}^{-1}\sigma_{2}^{-1}$
$\displaystyle=\sigma_{2}^{-1}(\sigma_{1}\sigma_{2}\sigma_{1})\sigma_{1}^{-1}\sigma_{2}^{-1}=\sigma_{2}^{-1}\sigma_{1}.$
(4)

A simple model Hamiltonian carrying these braids is

$H_{B}=\matrixquantity(1-e^{ik_{x}}&-i(1+e^{ik_{x}})&0\\\
-i(1+e^{ik_{x}})&e^{ik_{x}}-e^{ik_{y}}&-i(1+e^{ik_{y}})\\\
0&-i(1+e^{ik_{y}})&-1+e^{ik_{y}}),$ (5)

for $k_{x},k_{y}$ periodic, parametrized on $[-\pi,\pi]$. It represents the
momentum-space Bloch Hamiltonian of a real space model with nearest-neighbor
hoppings on a two-dimensional square lattice, where each unit cell contains
three orbitals. The spectrum along $(k_{x},-\pi)$ corresponds to
$a=\sigma_{1}$ , and to $b=\sigma_{2}$ along $(\pi,k_{y})$. As shown in Eq.
(4), these boundary braids do not commute. The total BZ boundary braid, along
the composition of the two great circles [Fig. 1(b)], is the nontrivial braid
$\sigma_{1}\sigma_{2}\sigma_{1}^{-1}\sigma_{2}^{-1}$. We show the spectrum on
the BZ boundary in Fig. 2 (b). In the supplemental material [44], we extend
the constructive method of Ref. [51] to show that there is always a way to
construct a given braid within a BZ-continuous model.

The entire spectrum of $H_{B}$ is shown in Fig. 2 (c). It hosts two EP2s
(which occur generically in two dimensions) that carry braid charges
$\sigma_{2}^{-1}$ and $\sigma_{1}$, compensating the boundary braid according
to the sum rule (2) or (4) explicitly. While they have opposite
Abelianization, their braid charges are not mutual inverses: These EPs are
_unpaired_.

Figure 3:  Non-Hermitian phase transition. (a) To gap out the system, any
contractible loop (fuchsia) in the BZ must carry a trivial braid. The only way
to change nontrivial braids (dashed) into trivial ones (dotted) continuously
is to move an EP across the loop. (b) Schematic representation of the spectrum
of Eq. (6) at $\delta=1$. The single nontrivial EP3 is marked in red, and the
green lines illustrate the real-gap closings which form nontrivial loops
around the BZ. (c) Schematic representation of the spectrum for the
interpolation Eq. (7) between the unpaired model hosting unpaired EPs and a
gapped phase. Initially the BZ boundaries (solid black lines) carry braid
charges $\sigma_{1}$ ($\sigma_{2}$) in the $k_{x}$ ($k_{y}$) direction. At
$s=3/4$ the two EPs merge at the boundary, breaking the $k_{x}$ braid charge
into a trivial braid; the corresponding boundary is then marked with dotted
lines. For $s>3/4$, all braids along $k_{y}=\textrm{const}$ are trivial, while
loops along $k_{x}=\text{const}$ correspond to braids of type $\sigma_{2}$.
The boundary braid in that regime is then
$\sigma_{2}\sigma_{2}^{-1}=\mathds{1}$. (d) Same as (c) for interpolation Eq.
(8) between nontrivial and trivial gapped phases for increasing interpolation
parameter $t$. A single twofold degeneracy is created at $t=2/3$ and
subsequently splits into two EP2s. These merge at the boundary for $t=6/7$.
For $t>6/7$ the system is topologically trivial and gapped; all loops in this
system correspond to the trivial braid. Note that the last panel in (c) and
the first panel in (d) describe the same system.

To illustrate this further, we tune this Hamiltonian to

$H^{\text{EP3}}_{\delta}=H_{B}+\delta\,2\sqrt{2}\operatorname{diag}(1,0,-1),\quad
0\leq\delta\leq 1,$ (6)

which leaves the boundary braids unchanged but merges the EP2s as $\delta\to
1$. We show the corresponding spectrum in Fig. 2(d). As the EP2s are unpaired,
they cannot annihilate, but fuse to a _single_ EP3 at $k_{x}=k_{y}=0$.

While fine tuned, this EP3 is significantly distinct from the simple third-
root EPs commonly implemented in topologically trivial spaces [23, 49, 33].
First, it can exist alone under the BZ periodic boundary condition. Second,
its dispersion consists of the roots of a third-order polynomial which forbids
a simple expression. Third, it lies at the intersection of two Fermi arcs,
along which one of the real gaps $\operatorname{Re}(E_{i}-E_{j})$ closes.
These Fermi arcs extend through the BZ, roughly following $(k_{x},0)$, and
$(0,k_{y})$. In particular they form extended noncontractible loops, unlike
the common line-segment Fermi arcs which are known to appear in topologically
trivial models.

_Topological robustness of the gapless phase._ The topological robustness of
the EP phase arises from the nontrivial braids along contractible loops on the
BZ torus, especially the loop successively traveling as $aba^{-1}b^{-1}$. We
show first that all contractible loops in a gapped area must carry trivial
braids: We take a deformation retraction $\gamma_{s}$ ($0\leq s\leq 1$)
between some loop $\gamma_{0}$ in the BZ and a constant loop $\gamma_{1}$ (a
point). If the Hamiltonian is gapped throughout this deformation retraction,
then there is a well-defined braid of eigenvalues $[\gamma_{s}]$ corresponding
to each $\gamma_{s}$, defining a continuous deformation of braids. Such
continuous deformation without intersecting strands does not affect the braid
group element, which means $[\gamma_{0}]=[\gamma_{1}]$. As the constant loop
carries the trivial braid this concludes the proof. This implies conversely
that if any contractible loop in the BZ carries a nontrivial braid, the system
must be gapless.

This argument implies robustness of the gaplessness under perturbation. In the
BZ of $H_{B}$, all contractible loops enclosing either or both of the EPs
carry nontrivial braids. The only way to break the braid along a loop is to
move EPs across it [Fig. 3(a)]. When perturbing $H_{B}$, the braids along
loops far away from the EP are unaffected. They are of the same type as
labeled by the great circles $aba^{-1}b^{-1}$. So the system remains gapless
and contains multiple EP2s or a single EP3, whose braids combine to
$\sigma^{-1}_{2}\sigma_{1}$.

It is instructive to consider how the degeneracies evolve when we transition
from the gapless phase to a gapped phase. We interpolate between Eq. (5) and a
model $H_{B}(\pi,k_{y})$ without $k_{x}$ dependence,

$H_{s}(k_{x},k_{y})=(1-s)H_{B}(k_{x},k_{y})+sH_{B}(\pi,k_{y}).$ (7)

Physically this corresponds to decoupling the unit cells in the $x$ direction.
The results are summarized in Fig. 3(c). As $s$ increases, the two EPs move
away from each other through the BZ towards $(0,\pm\pi)$. Their combined
trajectory traces over the meridian along a large nontrivial loop involving
the boundary. Finally, for $s=3/4$ they fuse at $(0,\pi)$. Further increasing
$s$ annihilates the two EPs and we obtain a gapped phase.

We may also understand this process and the robustness of the phase via the
Fermi arcs. The initial model hosts two topologically nontrivial Fermi arcs,
one along each great circle. These arcs can only end in degeneracies, and they
change continuously under continuous deformation of $H_{B}$. By removing
dependence on $k_{x}$, we remove the Fermi arc that follows $k_{y}$,
correspondingly, two degeneracies must follow the meridian around the torus.

The $s=1$ phase, while nondegenerate, is still a topologically nontrivial
phase. One of the three bands is effectively decoupled from the others, but
the other two are braided in $k_{y}$ direction. They share a corresponding
extended Fermi arc of $\operatorname{Re}(E_{2}-E_{3})$ closing along
$(k_{x},0)$. The spectrum along the BZ boundary corresponds to
$(a,b)=(\mathds{1},\sigma_{2})$. These two braid elements commute and the
combined braid along the boundary is trivial, consistent with previous
classification [25, 26]. We also present how this topologically nontrivial
gapped phase can be switched into a fully topologically trivial phase
$H_{0}=[2+\cos(k_{x})]\mathrm{diag}(1,0,-1)$. We choose the following
interpolation,

$H_{t}=(1-t)H_{B}(\pi,k_{y})+tH_{0}.$ (8)

The evolution of the spectrum is shown in Fig. 3 (d). At $t=2/3$, a new
degeneracy is created at $\mathbf{k}=(0,0)$. It splits into a pair of EP2s for
increasing $t$, which then separate further in the $k_{x}$ direction. Their
trajectory traces over a complete longitude, until they merge and annihilate
into the trivial gapped phase at $t=6/7$.

The above series of figures showcases how the topology of EP-annihilation
trajectories corresponds to different phase transitions. In the $H_{s}$
interpolation (7), the EP2s travel over the meridian to cross all contractible
loops enclosing the EP3 and break braid $a$. In the Supplemental Material
[44], we prove that if the two EP2s trace out a simple loop, it must be in the
noncontractible homotopy class, based on the universal covering map
$\mathbb{R}^{2}\to T^{2}$ and the Jordan-Schoenflies theorem [52, 53]. In the
$H_{t}$ interpolation (8), the newly created EPs must cross all meridians in
order to render the $b$ braid trivial.

_Conclusion._ We have constructed non-Hermitian phases with unpaired
exceptional degeneracies and explained their topological robustness,
illustrated in a short-range tight-binding lattice model.

The non-Abelian nature of three or more complex energy bands allows for
overall nontrivial topological charges and even non-Abelian monopoles under
periodic boundary conditions in reciprocal space. In nontrivial phases these
charges are unpaired and cannot be annihilated locally by perturbations.
Instead they must travel nontrivially around the BZ torus in order to be
gapped out. This endows the gapless phases presented here with topological
robustness, as the movement of degeneracies generally is continuous when
deforming a model.

Conversely, topologically inequivalent ways of moving EPs around the BZ lead
to distinct gapped and gapless phases. The movement of EPs thus encodes the
topological information of the system. This differs from the situation on a
plane [54], where at least three EPs are required to exhibit non-Abelian
features.

Under some tuning, our model exhibits an isolated nontrivial higher-order
degeneracy with dispersion beyond the $n$th-root behaviours constructed
previously in tight-binding models. This EP3 is accompanied by extensive Fermi
arcs that form noncontractible loops, instead of open line segments. These
topological Fermi arcs do not have counterparts in Hermitian Weyl nodes, and
are purely a feature of non-Hermitian systems.

_Note added_ : After the first version of our work, two experimental works
[55, 56] realized non-Hermitian multiband systems with nontrivial energy
braids in acoustic metamaterials, which corroborates the direct experimental
relevance of our work.

###### Acknowledgements.

Acknowledgments. We thank Lukas Rødland, Marcus Stålhammar and Zhi Li for
stimulating discussions. J.L.K.K., K.Y. and E.J.B. were supported by the
Swedish Research Council (VR, Grant No. 2018-00313), the Wallenberg Academy
Fellows program (2018.0460) and the project Dynamic Quantum Matter (2019.0068)
of the Knut and Alice Wallenberg Foundation, as well as the Göran Gustafsson
Foundation for Research in Natural Sciences and Medicine. K.Y. is also
supported by the ANR-DFG project (TWISTGRAPH).

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##  I Supplemental Material

###  I.1 Braid groups and exceptional points

The tour around the EP gives rise to a map, denoted as $\gamma$, from a circle
$S^{1}$ to the space spanned by gapped NH Hamiltonians
$\mathcal{W}_{N}=[\mathrm{Conf}_{N}(\mathbb{C})\times
GL_{N}(\mathbb{C})/(\mathbb{C}^{\times})^{N}]/S_{N}$ [25, 26]. The group
$GL_{N}(\mathbb{C})$ represents the general linear transformation to the
eigenbasis, from which we quotient out the transformations that give identical
diagonalization. These are scaling transformations
$\mathrm{diag}(z_{1},z_{2},\dots,z_{N})$ with $z_{j}\neq 0$, represented by
$(\mathbb{C}^{\times})^{N}=(\mathbb{C}\setminus\\{0\\})^{N}$. The
configuration space $\mathrm{Conf}_{N}(\mathbb{C})$ corresponds to the energy
spectrum, which is formed by lists of $N$ distinct complex numbers
$\\{E_{1},E_{2}\dots,E_{N}\\}$. The group $S_{N}$ consists of joint
permutations of eigenvalues and eigenvectors. The topology of this map can be
described by the homotopy group $\pi_{1}[\mathcal{W}_{N}]=B_{N}$ [25, 26],
which is the braid group on $N$ strands.

###  I.2 Base-point homotopy vs free homotopy descriptions

The topology of a point degeneracy or defect in two dimensions is described by
how the gapped Hamiltonian or the order parameter evolves along a loop around
it. There are two different but mutually complementary ways of describing it,
the base-point homotopy theory and the free homotopy theory. The former has
the advantage of describing operations on the degeneracy (defects) through the
powerful group theory, while it needs to artificially assign a base point
somewhere. The latter is independent of such choice and more natural in
intuition. However, the free homotopy theory does not have a group structure
and is not appropriate when studying what kind of defects or degeneracy can
merge or split. This is the reason why we use the base-point homotopy in the
main text. In this section, we introduce its connection to the freely
homotopic classes.

The base-point homotopy group $\pi_{1}(X,x_{0})$ is defined with respect to
all maps $\gamma$ from $S^{1}$ to some space $X$ with a fixed point in $S^{1}$
mapped to a fixed point $x_{0}\in X$. To make the language clearer, we
identify $S^{1}$ as the unit interval $[0,1]$ with periodic boundary
condition. The base-point homotopy group is to study all continuous maps
$\gamma:[0,1]\to X$ satisfying $\gamma(0)=\gamma(1)=x_{0}$. The point $x_{0}$
is called the base point. The base point plays an important role in
constructing a group structure, such that different loops can be composed from
the common base point. The composition of two loops
$\gamma_{1}\cdot\gamma_{2}$ is regarded as a loop that first travels from
$x_{0}$ along $\gamma_{1}$ to $x_{0}$, and then do the second leg along
$\gamma_{2}$. Two maps $\gamma_{1}$ and $\gamma_{2}$ starting at $x_{0}$ are
homotopic if they can be interpolated continuously by a series of loops from
$x_{0}$ (Fig. S1). The homotopic class of $\gamma$ is denoted by $[\gamma]$.

For a path-connected space, one can show that the homotopy groups
$\pi_{1}(X,x_{0})$ at different base points only differ by an isomorphism.
Taking two base point $x_{0}$ and $x^{\prime}_{0}$, as the space is path-
connected, we can find a path $\gamma_{0}$ with joints them. A loop $\gamma$
starting from $x_{0}$ can be extended to a loop from $x^{\prime}_{0}$ by
$\gamma_{0}\cdot\gamma\cdot\gamma^{-1}_{0}$. So $\gamma_{1}\cdot\gamma_{2}$ is
mapped to
$\gamma_{0}\cdot\gamma_{1}\cdot\gamma^{-1}_{0}\cdot\gamma_{0}\cdot\gamma_{2}\cdot\gamma^{-1}_{0}$.
Notice that the loop $\gamma^{-1}_{0}\cdot\gamma_{0}$ is homotopic to a
constant loop. So we have $[\gamma^{-1}_{0}]\cdot[\gamma_{0}]=\mathds{1}$.
When taking the homotopic classes, this correspondence becomes
$[\gamma_{1}]\cdot[\gamma_{2}]\to[\gamma_{0}]\cdot[\gamma_{1}]\cdot[\gamma_{2}]\cdot[\gamma^{-1}_{0}]$,
preserving the group product. The inverse homomorphism from
$\pi_{1}(X,x^{\prime}_{0})$ to $\pi_{1}(X,x_{0})$ is given by the action of
$\gamma_{0}^{-1}$. Thus $\pi_{1}(X,x^{\prime}_{0})$ and $\pi_{1}(X,x_{0})$ are
isomorphic. Under such identifications, one may not write the base point out
explicitly and use $\pi_{1}(X)$ to denote the abstract structure of the base-
point homotopy group.

Now we introduce the freely homotopic class $[S^{1},X]$. In this definition,
two loops $\gamma_{1}$ and $\gamma_{2}$ belong to the same class if they can
be interpolated continuously by a series of loops in $X$, without requiring a
fixed starting point (see Fig. S1). Compared to the base-point homotopy, the
free homotopy allow more ways of identifying loops. And it characterizes the
defects in a more invariant way as there is no artificial choice of
references. However, due to the lack of a base point, there is no group
structure in $[S^{1},X]$. It is not a good tool when talking about the
composition or separation of degeneracy.

In order to see the connection between the two homotopic classes, one can
consider a loop $\gamma^{\prime}$ from $x_{0}$ to itself. Its action on
$\pi_{1}(X,x_{0})$ is an automorphism of the base-point homotopy:
$[\gamma^{\prime}]\cdot\pi_{1}(X,x_{0})\cdot[\gamma^{\prime-1}]$. This
operation can be considered as a free homotopy which moves the base point
$x_{0}$ along $\gamma^{\prime}$ [47]. So this is a map from the base-point
homotopic classes to the free homotopic classes. One can further prove that
this is indeed a bijection map. The freely homotopy classes can be identified
by [47]

$\pi_{1}(X,x_{0})/\pi_{1}(X,x_{0})\simeq[S^{1},X],$ (S1)

where the quotient should be understood as the conjugate action introduced.

In the base-point homotopy language, the EP is characterized by the braid
group element $[\gamma]$. From the above discussion, if we choose a different
base point but the same loop, the corresponding braid group element becomes
$[\gamma_{0}]\cdot[\gamma]\cdot[\gamma_{0}^{-1}]$. So in the base-point
homotopy description, the correspondence between braid elements and the
topological types of EP is not unique. Braid elements conjugated to each other
can be equivalent EPs. This ambiguity is resolved in the freely homotopic
description. For the case of non-Hermitian EPs, the classes in
$[S^{1},\mathcal{W}_{N}]$ can be regarded as gluing the start configuration
and end configuration of a braid together. It is a actually a knot embedded in
a solid torus [25, 27, 46].

Figure S1: Left: the definition of base-point homotopic loops and freely
homotopic loops and the isomorphism between the homotopy groups with respect
to different base points. Right: When a loop is contractible to a point
$x_{f}$, it is also contractible when a base point is fixed. This can be shown
be combining the contraction with a path connecting the base point to the
point $x_{f}$.

To conclude this section, we prove the following statement used in the main
text:

  * •

_If a loop $\gamma:S^{1}\to X$ is contractible, it is also contractible when a
base point is fixed. That is, a trivial element in the freely homotopic class
$[S^{1},X]$ is also trivial in the base-point homotopic class $\pi_{1}(X)$._

Although this statement may be observed immediately from the relation (S1), we
here give a detail construction for the equivalence (also Fig. S1), in order
to complement the discussion of braid protection in the main text. As before,
we treat $S^{1}$ as the unit interval $[0,1]$ with periodic boundary
condition. The loop is written as map $\gamma:[0,1]\to X$. The contraction is
a map $\alpha:S^{1}\times[0,1]\to X$ with
$\alpha(s,0)=\gamma(s),\alpha(s,1)=x_{f}$. Without loss of generality, we take
the point $x_{i}=\gamma(0)$ as the base point. We show that we can find a map
$\beta:S^{1}\times[0,1]\to X$ such that
$\beta(s,0)=\gamma(s),\beta(s,1)=x_{i}$ and $\beta(0,t)=\beta(1,t)=x_{i}$.

During the contraction $\alpha$, the point $x_{i}$ actually trace out a path
from $x_{i}$ to $x_{f}$. We use $\gamma^{\prime}$ to represent this path. For
each $t$, we notice that $\alpha(S^{1},t)$ is a loop starting from
$\gamma^{\prime}(t)$. We label the path from $\gamma^{\prime}(0)$ to
$\gamma^{\prime}(t)$ as $\gamma_{t}$. Then
$\gamma_{t}\cdot\alpha(S^{1},s)\cdot\gamma_{t}^{-1}$ is a loop starting from
$x_{0}$, traveling to $\gamma^{\prime}(t)$, doing a loop along
$\alpha(S^{1},t)$ and traveling back to $x_{0}$. By doing this tour for every
$t$, we construct a series loops $\alpha^{\prime}:S^{1}\times[0,1]\to X$ with
$\alpha^{\prime}(s,0)=\gamma(s),\alpha^{\prime}(s,1)=\gamma^{\prime}\cdot\gamma^{\prime-1}(s)$
and $\alpha^{\prime}(0,t)=\alpha^{\prime}(1,t)=x_{i}$. This is a base-point
homotopy between $\gamma$ and the composition path
$\gamma^{\prime}\cdot\gamma^{\prime-1}$. The latter obviously is homotopic to
a constant map at $x_{0}$ via some map $\alpha^{\prime\prime}$. So the desired
base-point homotopy from $\gamma$ to a constant map is given by
$\beta=\alpha^{\prime}\cdot\alpha^{\prime\prime}$.

###  I.3 Trivial single EPs of any order

We first present single EP2s corresponding to trivial eigenvalue braids.
Consider

$H=\begin{pmatrix}0&1\\\ f(\mathbf{k})&0\end{pmatrix}$ (S2)

which has eigenvalues $E=\pm\sqrt{f(\mathbf{k})}$ and an exceptional point at
$f(\mathbf{k})=0$. With $f(\mathbf{k})=2-e^{ik_{x}}-e^{ik_{y}}$, we have a
single EP at $\mathbf{k}=0$ in a periodic two-dimensional model. Near the EP,
we have $f(\mathbf{k})\simeq-i(k_{x}+k_{y})+|k|^{2}/2$, thus the dispersion is
square root in generic directions, whereas it is linear only along the high-
symmetry line $k_{x}=-k_{y}$ along which the linear terms cancel.

This degeneracy is however only stable to particular perturbations. Any
perturbation around the EP can be represented by replacing $f(\mathbf{k})$ by
$f(\mathbf{k})+\epsilon$ [57, 50]. If we want to keep the system gapless,
there should exist a solution to $\epsilon-i(k_{x}+k_{y})+|k|^{2}/2=0$. So we
must have $\textrm{Re }\epsilon=-|k|^{2}/2$ and $\textrm{Im
}\epsilon=(k_{x}+k_{y})$. The second equation can always be satisfied while
the first equation imposes restrictions on $\epsilon$. The EP is therefore
gapped out if $\textrm{Re }\epsilon>-|\textrm{Im}\epsilon|^{2}$, while
splitting into pairs of nontrivial EP2s for $\textrm{Re
}\epsilon<-|\textrm{Im}\epsilon|^{2}$.

We show how trivial EPs can be generalized to more situations. For the
$2\times 2$ matrices with $1$ and $f(\mathbf{k})$ on the off diagonal, there
is a single EP at $\mathbf{k}=0$ with the choice in $d$ dimensions

$f(\mathbf{k})=d-\sum_{j=1}^{d}e^{ik_{j}}.$ (S3)

This gives a single EP2 at ($\mathbf{k}=0$) in a periodic model in any
dimension $d$. Again the dispersion is square-root in generic directions
whereas it is linear along high-symmetry lines along which the linear terms
cancel, i.e. $\sum_{\\{j\\}}k_{j}=0$ summed over some (sub)set of momenta.

The scenario above for second order EPs generalize directly to any order. To
this end consider the $n\times n$ matrix

$H_{n}=\begin{pmatrix}0&1&&&&&&&\\\ 0&0&1&&&&&&\\\ &0&0&1&&&&&\\\
&&0&0&&&&&\\\ &&&&&\ddots&&&\\\ &&&&&&0&1&\\\ &&&&&&0&0&1\\\
f(\mathbf{k})&&&&&&&0&0\end{pmatrix},$ (S4)

which has eigenvalues fulfilling $(-E)^{n}=f(\mathbf{k})$ hence
$E\propto\sqrt[n]{f(\mathbf{k})}$. At $f(\mathbf{k})=0$ there is an $n$-th
order EP, i.e. an EP$n$. Here we note that with the choice of $f$ in Eq. (S3)
the dispersion is generically $\sim k_{1}^{1/n}$ and in high symmetry
directions maximally $\sim k_{1}^{2/n}$, hence it has the telltale EP feature
of a sublinear dispersion in all directions. Of course it also has a single
right (and left) eigenvector.

###  I.4 Construction of unpaired degeneracies on a torus from given braids

We recapitulate the construction of a Hamiltonian that has a point degeneracy
with some specific braid $B$ as a topological invariant, as presented in Ref.s
[27, 46]. This builds on the mathematical construction in Ref. [51] of a
polynomial with a given braid as its root set. We show that a continuous
extension to the torus is possible whenever the braid $B$ is a commutator,
albeit this may require infinite hopping range, not as simple as the model in
the main text.

The construction starts from building a Hamiltonian that gives the desired
braid in the vicinity of the EP. We begins with the following ansatz

$H_{c}(k)=\matrixquantity(-c_{M-1}(k)&&\cdots&&-c_{0}(k)\\\
1&0&\cdots&\cdots&0\\\ 0&\ddots&\ddots&&\vdots\\\
\vdots&\ddots&\ddots&\ddots&\vdots\\\ 0&\cdots&0&1&0\\\ ,)$ (S5)

which has as its eigenvalues the roots $\lambda(k)$ of

$p(\lambda,k)=\lambda^{M}+\sum_{j=0}^{M-1}c_{j}(k)\lambda^{j}.$ (S6)

We set $\mathbf{k}=0$ as the EP position. The goal is then converted to
constructing a characteristic polynomial with the appropriate roots. This
reduces further to finding a simple expression for the roots as complex
functions $Z_{i}(k)$, since then the polynomial

$p=\prod_{j=1}^{M}\left(\lambda-Z_{j}(k)\right)$ (S7)

has the desired root set.

However, finding an expression for the roots is nontrivial, since they are
multi-valued functions whenever the braid $B$ considered is not pure.

Our construction proceeds as follows. Following Ref. [51], we construct the
polynomial $p(k=\exp(i\phi))$ on a circle, by a truncation of the Fourier
expansion of the braid diagram. We then extend it to a disk
$\set{z\in\mathbb{C}}{\absolutevalue{z}<=1}$ with this circle on the boundary,
such that the corresponding Hamiltonian is degenerate at a single point,
following Ref. [27]. Finally we embed this disk in a torus, defining $p$ on
$T^{2}$.

Assume without loss of generality that $B$ consists of only one cyclic
permutation cycle of length $M$, i.e. following strand $1$, we end at the
starting point of strand $2$, and so does this to strand $2$ and $3$, until
some strand $M$ that ends in the starting point of strand $1$. Models
containing multiple such cycles can be obtained as tensor products of the one-
cycle case or by multiplying their corresponding polynomials $p$. Some care
must be taken to avoid degeneracies between the constituent components.

The braid can then be represented by its braid diagram, which embeds into the
complex numbers as $M$ piecewise linear strands. We choose here to represent
the ordering of the $M$ strands along the real line, and overcrossing,
respectively undercrossing, of strands by the difference in imaginary parts at
the points where the real parts cross.

Since the braid $B$ consists of one $M$-cycle by assumption, the traversal of
all $M$ strands in succession is periodic. The braid can thus be represented
by a single $2\pi$-periodic function $f$, that follows strand $i$ on the $i$th
interval of length $2\pi/M$ in its domain.

This function $f$ is chosen to be piecewise linear, and thus at least
$1$-Hölder. Hence there is a triangular-polynomial approximation $S_{N}f$ that
converges to it uniformly,

$\forall\phi\in[0,2\pi]:\
\absolutevalue{\Delta_{N}(\phi)}=\absolutevalue{(S_{N}f-f)(\phi)}\leq
C_{1}\ln(N)/N.$ (S8)

We can therefore approximate the desired polynomial $p$ using this $S_{N}f$,
via

$p_{N}(\phi,z)=\prod_{s=1}^{M}\left(z-S_{N}f\left(\frac{\phi}{M}+\frac{2\pi
s}{M}\right)\right)$ (S9)

This polynomial is well-defined, since the terms are simply permuted $s\to
s+1\mod M$ as $\phi\to\phi+2\pi$.

We need to take care that no two factors in this product are the same, to keep
the resulting Hamiltonian nondegenerate. This is the case if we either
manually post-process $S_{N}$ as is done in Ref.[51], but choosing
sufficiently large $N$ is going to satisfy this automatically since any given
braid has a finite minimal gap, and choosing large $N$ keeps $S_{N}f$ close to
$f$ s.t. there are no added crossings due to the uniform convergence we show
in the next subsection.

Even though the Fourier series is evaluated for $\phi/M$, the authors show
that approximations of this type result in polynomial dependence of $p_{N}$on
$\exp(\pm i\phi)$, i.e. $p_{N}=p_{N}(\lambda,\exp(i\phi))$. As shown in Ref.
[46], this polynomial can be extended to the unit disk in the complex plane by
replacing $\exp(i\phi)=z/\absolutevalue{z}$, and possibly regularising to the
polynomial

$\tilde{p}_{N}(\lambda,z)=\absolutevalue{z}^{g*M}p_{N}\left(\lambda/\absolutevalue{z}^{2g},z/\absolutevalue{z}\right)$
(S10)

with some integer $g>0$. The extension to the disk is done for some
sufficiently large but fixed $N=N_{0}$.

Next, we show that this model can be continuously extended from the unit disk
to the torus. Note first that $p_{N}\to p$ uniformly in $\phi$, since

$\displaystyle P_{N}(z,\phi)$
$\displaystyle=\prod_{s=1}^{M}\left(z-S_{N}f\left(\frac{\phi}{M}+\frac{2\pi
s}{M}\right)\right)$ (S11)
$\displaystyle=\prod_{s=1}^{M}\Bigg{(}z-f(\ldots)+\underbrace{f(\ldots)-S_{N}f\left(\frac{\phi}{M}+\frac{2\pi
s}{M}\right)}_{=:\Delta_{N}^{[s]}\left(\phi\right)}\Bigg{)}$ (S12)
$\displaystyle=P(z,\phi)+\sum_{m=0}^{M-1}\Delta_{N}^{[m]}(\phi)\prod_{s=m+1}^{M-1}\left(z-f\left(\frac{\phi}{M}+\frac{2\pi
s}{M}\right)\right)$ (S13)

by partial expansion of the product. Thus we obtain the bound

$\absolutevalue{P_{N}-P}\leq MC_{1}C_{2}^{M}\ln(N)/N$ (S14)

with $C_{2}=\max_{\phi}\absolutevalue{z-f(\phi)}$ from the uniform convergence
of $S_{N}f\to f$ in Eq. (S8).

To extend the model to the torus, we simply restore the higher order terms
$S_{N}f,N>N_{0}$ to the triangular-polynomial approximation s.t. at the
boundary of the chosen torus unit cell we have the model defined by $f$,
respectively $p$. This means that the eigenvalues of the Hamiltonian on this
boundary are precisely the piecewise linear eigenvalues defined by the braid
diagram. For a braid $B$ that is a commutator, this has the required symmetry
on the boundary, and is thus a well-defined map on the torus.

The constructive procedure above guarantees a BZ-continuous model. However, in
general it does not lead to a differentiable or analytic models in
$\mathbf{k}$. This means infinite-range hopping terms may be required.

###  I.5 Topological robustness of the EP3 or pair of EP2

_Annihilation condition_ : According to the discussion in the main text, after
the EP$3$ split into two EPs under perturbation, these two EPs cannot be
directly brought together to be gapped out, since any contractible loop
enclosing them carries a nontrivial braid. We denote the location of the EP3
as $x_{0}$ and a small disc containing it as $D_{0}$ (Fig. S2). In order to
break these braids protecting the degeneracy, the two EPs must trace over some
large loop $C$ and cross all loops homotopic to $\partial D_{0}$ on
$T^{2}-\\{x_{0}\\}$. After this they can be annihilated.

Figure S2: If the two EPs trace out a simple loop $C$ that is contractible on
the torus, we can lift it to the covering space $\mathbb{R}^{2}$. There we can
always construct a loop homotopic to the circle around the EP, while it does
not intersect with the trajectory of EP, using the tubular neighborhood of
$C$. So the nontrivial braid along it is not broken. Thus the loop $C$ must be
some nontrivial loops on the torus.

_Topology of the EP trajectory_ : In order to study the topological property
of the curve $C$, we may assume that one EP is fixed and $C$ is traced over by
the other EP. Then $C$ is regarded as a loop starting and ending at $x_{0}$.
An immediate consequence is that $C$ must either intersect with all meridians
or all longitudes. Otherwise we can take the nonintersected meridian and
longitude as the boundary of the BZ. There is obviously a loop homotopic to
$\partial D_{0}$ that does not cross $C$.

First, we prove that if $C$ is a simple loop that does not have any self-
intersection, it must belong to the nontrivial classes of
$\pi_{1}(T^{2},x_{0})$. Notice here we are interested in the image of $C$
instead of the map $C:S^{1}\to T^{2}$ itself. Only in the simple loop
situation is the answer directly related to the homotopy group
$\pi_{1}(T^{2},x_{0})$.

To make the question easier, we use the universal covering map
$p:\mathbb{R}^{2}\to T^{2}$ to lift $C$ from the BZ to the two-dimensional
plane. The advantage of lift is that we have a clear definition for the
interior and the exterior of a simple loop. And we choose
$p[(2m\pi,2n\pi)]=x_{0}$. We look for contradiction if $C$ is a contractible
loop. If $C$ belongs to the trivial class of $\pi_{1}(T^{2},x_{0})$, it has a
unique lift $\tilde{C}$ starting and ending at $(0,0)$ [52], which is also a
loop. Since $C$ does not intersect with itself, either does $\tilde{C}$. The
preimages $p^{-1}(C)$ are made up of all translations of $\tilde{C}$.
Moreover, the translations of $\tilde{C}$ do not intersect with $\tilde{C}$
since $C$ is a simple loop. So there is always a neighborhood of $\tilde{C}$
that contains no other preimages of $p^{-1}(C)$. According to the
Jordan–Schoenflies theorem [53], $\tilde{C}$ separates $\mathbb{R}^{2}$ into
two connected components $W,W^{\prime}$, with $\tilde{C}$ their common
boundary. The enclosure of the bounded component (interior) $\bar{W}$ is
actually homeomorphic to a closed disc. We can prove that $\bar{W}$ does not
contain any points of $p^{-1}(x_{0})$ except for $(0,0)$. If not so, $\bar{W}$
contains another $y\in p^{-1}(x_{0})$, the lift $\tilde{C}^{\prime}$ of $C$
starting from $y$ is a connected space on $\mathbb{R}^{2}-\tilde{C}$. So it
must lie in $\bar{W}$. However this is not possible, since
$\tilde{C}^{\prime}$ is a translation of $\tilde{C}$.

We assume that $\tilde{C}$ is an embedding of $S^{1}$ in $\mathbb{R}^{2}$. It
has a tubular neighborhood $N$ that is diffeomorphic to its normal bundle
[47]. This normal (line) bundle is trivial as it is orientable (from the
orientability of $\tilde{C}$ and $\mathbb{R}^{2}$). So it can be taken as
$\tilde{C}\times(-1,1)$ with $\tilde{C}\times 0$ identified with $\tilde{C}$.
The connected component $\tilde{C}\times(0,1)$ may either lie in $W$ or
$W^{\prime}$ and similarly for $\tilde{C}\times(-1,0)$. Without loss of
generality, we choose $\tilde{C}\times(0,1)\subset W^{\prime}$. Now we can
construct a loop homotopic to $\partial D_{0}$ with no cross with $p^{-1}(C)$.
We may simply take $D_{0}\in N$ around $(0,0)$, whose boundary intersects
$\tilde{C}$ at $t_{1},t_{2}$. Then we choose two points
$t^{\prime}_{1},t^{\prime}_{2}\in\partial D_{0}\cap W^{\prime}$ near $t_{1}$
and $t_{2}$ respectively. The small arc connecting $t^{\prime}_{1}$ and
$t^{\prime}_{2}$ is taken to the small arc on $\partial D_{0}$, while the
large arc connecting them is taken to be a curve nearly parallel to
$\tilde{C}$. We denote this loops as $l_{t^{\prime}_{1},t^{\prime}_{2}}$. It
is homotopic to $(\partial
D_{0}\cap\bar{W}^{\prime})\cup[\tilde{C}\cap(\mathbb{R}^{2}-D_{0})]$. The
second part $[\tilde{C}\cap(\mathbb{R}^{2}-D_{2})]$ is homotopic to $\partial
D_{0}\cap\bar{W}$ since $\bar{W}-D_{0}$ is simply connected. By projecting
back the loop $l_{t^{\prime}_{1},t^{\prime}_{2}}$ to the torus, we obtain a
loop that is homotopic to $\partial D_{0}$ on $T^{2}-{x_{0}}$ while its braid
is not destroyed by the trajectory of EPs. This does not meet the annihilation
condition. So a simple loop $C$ of EP trajectory must lie in the nontrivial
classes of $\pi_{1}(T^{2},x_{0})$, which are generated by meridians and
longitudes.

We show that for trajectories $C$ with finite self-interactions, the image of
$C$ should contain great circles of the torus. However, in this situation, a
contractible loop like the one along the BZ boundary also meets the criterion
of crossing all contractible loops in $\pi(T^{2},x_{0})$. If the loop contains
finite number of self-crosses, then we parameterize the loop as a map
$c:S^{1}\to T^{2}$ on the interval $[0,2\pi)$. The self intersecting points
are ordered as $s_{0}<s_{1}<s_{3}\dots$. The total number of self. Then there
is a maximal $i$ such that $c(s_{i})=c(s_{0})$. If the loop $c[s_{0},s_{i}]$
contains any meridians or longitudes, we are done. If not, this loop is
contractible and we can replace it by a constant
$c^{\prime}[s_{0},s_{i}]=c(s_{0})$. Choose $c^{\prime}(s)=c(s)$ for
$s\not\in[s_{0},s_{1}]$, we obtain a new loop. Continue repeating this
examination until all self-intersecting points are sorted, we obtain a simple
contractible loop $C_{1}$. As before we lift this loop to the cover space
$\mathbb{R}^{2}$. The lifted loop $\tilde{C}$ does not intersect with its
translation counterparts, otherwise it must contain some meridians or
longitudes. The lifted loop $C_{1}$ separates $\mathbb{R}^{2}$ into two
connected components $W_{1},W^{\prime}_{1}$. Now examine
$\tilde{C}\cap\bar{W}^{\prime}_{1}$. This curve includes $\tilde{C}_{1}$ and
those segments intersecting $\tilde{C}_{1}$ finite times, at a subset
$\\{s^{\prime}_{j}\\}\subset\\{s_{j}\\}$. As before, we start to look at the
first intersecting point $s^{\prime}_{1}$. If $\tilde{C}$ travels along
$\tilde{C}_{1}$ after $s^{\prime}_{1}$ or move towards the bounded component
$W_{1}$, we do not need to do anything and look at the next intersecting
point. If $\tilde{C}$ moves towards the unbounded component $W^{\prime}_{1}$,
it will intersect again with $\tilde{C}_{1}$ at some point $s^{\prime}_{j}$.
First consider $c(s^{\prime}_{j})\neq c(s^{\prime}_{1})$. Denote this curve as
$w_{s^{\prime}_{1},s^{\prime}_{j}}$. It may contain some self-intersections,
which we can as before contract to a constant map. So
$w_{s^{\prime}_{1},s^{\prime}_{j}}$ can be treated as a simple curve. Together
with the two arcs on $C_{1}$ connecting $s_{1}$ and $s^{\prime}_{j}$,
$w_{s^{\prime}_{1},s^{\prime}_{j}}$ can form two simple loops. Each of them
separates $\mathbb{R}^{2}$ into two connected components. We choose the one
that intersects with the EP. We denote this simple loop as $\tilde{C}_{2}$. In
this way we eliminate all self-intersections of $\tilde{C}$ in the interior of
$\tilde{C}_{2}$. If $c(s^{\prime}_{1})=c(s^{\prime}_{j})$, we may slight
deform the curve around $s^{\prime}_{j}$ so that it does not affect other
intersections or connecting $\tilde{C}$ with its translations. Repeating doing
this until we obtain some simple loop $\tilde{C}_{j}$ whose exterior does not
intersect with $\tilde{C}$. Then we may as before construct a contractible
loop $l$ on $\mathbb{R}^{2}-p^{-1}(x_{0})$ that lies on the exterior of
$\tilde{C}$ while does not intersect with its translation counterparts. Notice
that $\tilde{C}_{j}$ is only piece-wise smooth. We might first construct
segments of $l$ in the tubular neighborhood of each smooth piece of
$\tilde{C}_{j}$. And then connect them together in the neighborhood of
nonsmooth points. Projecting back $l$ to the torus we obtain the conclusion.

###  I.6 Boundary Braid Illustration

To better illustrate the braid nature of the boundary spectra, we include an
alternative version of Fig. 2 (b), where we plot the spectrum as a three-
dimensional plot, see Fig. S3.

Figure S3: Three-dimensional plot of the boundary braid, alternative to Fig. 2
(b) in the main text. The color corresponds to the imaginary part of the
eigenvalues.