\section{Introduction}
\label{Sec:Introduction}

It has been long recognized that there is a reciprocal relation between an underlying pattern of coupled resonators and the collective dynamical processes supported by that pattern. It was Jean Bellissard who made this statement very precise, when he introduced $C^\ast$-algebras and their K-theories to the condensed matter community \cite{Bellissard1986}. His observation was that, in a Galilean invariant theory, the underlying physical pattern determines uniquely the algebra generated by all possible dynamical matrices. In its most general form, this algebra is the stabilization of a groupoid $C^\ast$-algebra \cite{KellendonkRMP95} and, in its simplest form, it is the stabilization of a group $C^\ast$-algebra.\footnote{Stabilization simply means tensoring with the algebra ${\mathbb K}$ of compact operators.} Now, if these stabilized algebras are found to be isomorphic for two patterns, then any dynamical feature observed for one pattern has a counterpart for the second pattern. In other words, from dynamical point of view, the two patterns are identical. In turn, this implies a certain relation between the underlying patterns. For example, if the patterns were generated by acting with space groups, as explained in section~\ref{Sec:Gen}, then we can be sure that the two patterns were generated with space groups that belong to the same isomorphism class. For generic patterns, one will concluded that the transversal of the patterns, as defined in \cite{KellendonkRMP95},\footnote{This transversal serves as the space of units for the groupoid we already mentioned.} have homeomorphic blowups \cite{WilliamsBook}[Th.~2.52]. In all cases, an isomorphism between the algebras of dynamical matrices implies that the patterns belong to a well defined class of patterns. When metamaterials are approached from this angle, the following simple but deep principle emerges: Discovering new dynamical patterns in metamaterials amounts to exploring different stably isomorphic\footnote{Stably isomorphic algebras are said to be Morita equivalent.} classes of $C^\ast$-algebras. The present work advances this program by exploring classes of patterns associated with the discrete groups of symmetries of a strip (see section~\ref{Sec:SymmetricPatterns}).

\vspace{0.1cm}

The operator algebraic approach with its K-theoretic tools introduced by Bellissard has much more to offer \cite{Bellissard1986,BellissardJMP1994,
Bellissard1995,Bellissard2000,Bellissard2003}. Indeed, stably isomorphic or Morita equivalent $C^\ast$-algebras have identical K-theories. Therefore, the K-theoretic groups can be regarded as topological invariants for the deformations of the patterns and of the dynamical matrices within one class. Thus, after restricting to a fix algebra of dynamical matrices, hence to a well defined class of patterns, each connected component of the spectrum of a dynamical matrix, aka spectral band, can be labeled by an element of the $K_0$-group of the algebra. This element represents the complete topological invariant associated to the corresponding spectral projection, called band projection from now on. Indeed, if the $K_0$-group elements are found to be identical for all band projections of two different dynamical matrices, then we can be sure that the two dynamical matrices can be stably deformed into each other, inside the fixed algebra, without changing the topology of the spectrum. However, if the labels are found to be different, then the topology of the spectrum must change during the deformations and some of the spectral gaps must close, regardless how hard we try to avoid this phenomenon. This supplies a very simple, robust and practical principle for resonant spectrum engineering of metamaterials because it supplies the means to close and open spectral gaps, hence to generate topological spectral flows, on demand.

\vspace{0.1cm}

To apply the above principle in practice, one needs to: 
\begin{enumerate}[\ \ \it N1.]
\item Identify the class of patterns; 
\item Compute its corresponding $C^\ast$-algebra; 
\item Compute the $K_0$-group of this algebra, together with an explicit basis; 
\item Supply the generating model dynamical matrices. 
\end{enumerate}
If all these four components are in place, then a theorist can advise an experimental research group on how to reconfigure a meta-material to produce robust spectral flows that close and open specific spectral gaps. We should emphasize that all four components are equally important. For example, for a metamaterial scientist, it is not enough to specify the algebra of dynamical matrices, {\it e.g.} via some constraints such as equivariance w.r.t. to a group of symmetries. Indeed, applied metamaterial scientists work directly with physical structures and they need to know what are the allowed physical deformations  for the predictions to hold. Hence, the class of patterns itself needs to be specified explicitly. Also, computing just the $K_0$-groups is not sufficient for practical applications, because one also needs an explicit representative from the algebra for each $K_0$-equivalence class. In other words, one needs to specify an explicit basis for the $K_0$-group, from which one can derive model dynamical matrices (see subsection~\ref{Sec:Why}). Furthermore, in a different scenario, one may be dealing with a complicated dynamical matrix coming from a laboratory, in which case one will be interested to resolve the complete topological invariant of the band projections. In this scenario, one needs:

\begin{enumerate}[\ \ \it N5.]
\item A practical algorithm to compute the $K$-theoretic labels. 
\end{enumerate} 

In this work, we resolve all the above for patterns generated from frieze groups \cite{ConwayBook}, together with a demonstration of the working principles in phonic meta-materials. Specifically: 1) We introduce an algorithm that generates symmetric patterns by acting with the space transformations of a frieze group on a set of seeding shapes. The specific class of patterns considered in this work consists of those generated from seeding shapes placed at generic configurations with trivial site symmetry groups. In fact, we exclude any seed configuration that leads to overlapping shapes when acted with the space transformations of a give frieze group. 2) If the shapes of such pattern are actual resonating structures, {\it e.g.} elastic plates, acoustic cavities or solid bodies, and if the interaction between a pair of two resonators is fully determined by their relative geometric configuration, as it is always the case if the physics involved is Galilean invariant, then we show that the dynamical matrices governing the collective dynamics of the resonators belong to the stabilized group $C^\ast$-algebra of the corresponding frieze group. 3-4) We supply a complete account of the K-theory of this algebra for all seven classes of frieze groups, together with explicit sets of generators of the $K_0$-groups, as well as model dynamical matrices. 5) For a generic dynamical matrix, we also supply a numerical algorithm for computing the complete set of K-theoretic labels of the band projectors. This algorithm never makes appeal to a Bloch decomposition since it is entirely developed inside the real-space representation. Among other things, we believe this approach can handle disordered perturbations. 6) Lastly, we exemplify the model dynamical matrices using phonic crystals and we use COMSOL\footnote{COMSOL Multiphysics is a widely adopted commercial platform for finite element analysis, known to produce reliable simulations of continuous mechanics models.} simulations to demonstrate various topological spectral flows generated with the principles discussed above. 

\vspace{0.2cm}

The frieze groups are the simplest among the Euclidean space groups and the latter have the simplest structure among the discrete groups. Indeed, the Euclidean space groups have a large commutative normal subgroup, the subgroup of translations, and the quotient by this subgroup results in a finite group, the point group. As we shall see already in this work, this feature simplifies many of the tasks, such as that of generating a complete sets of pairings returning a complete set of numerical topological invariants (see Sec.~\ref{Sec:NumInv}). The present work, however, can be seen as a model of analysis for a general program that seeks to generate interesting dynamics from group $C^\ast$-algebras, {\it e.g.} for generic groups of generators and relations \cite{CoxeterBook1980}. Such groups act naturally on their corresponding Cayley graphs and the dynamics generated on the $\ell^2$-space of the graph can be connected with processes occurring in materials self-assembly and biology. While these connections are yet to be formulated more precisely and generally, examples of such dynamics exhibiting topological features can be found in \cite{TangPRX2021}.

\vspace{0.1cm}

Lastly, we want to specify that, in this work, we will not enforce any of the fundamental symmetries, that is, the time-reversal, particle-hole or chiral symmetries. Still, we want to remind the reader that these symmetries are equally relevant for quantum and classical degrees of freedom. For example, a practical method to implement these symmetries and their signatures in passive metamaterials can be found in \cite{BarlasPRB2018}, which recently resulted in the first topological metamaterial from class D (see \cite{ChengArxiv2022}) with anti-unitary particle-hole symmetry. Therefore, we will certainly try to incorporate these symmetries in our future investigations and applications.

\vspace{0.2cm}

We now discuss the relation between our work and the existing literature. We start with a brief survey of the physics literature. The first indication that a space symmetry can enrich the topological landscape of a material appeared in \cite{FuPRL2011}. The first indication that a space symmetry {\it alone} can stabilize topological phases in a material came from Refs.~\cite{TurnerPRB2010,HughesPRB2011}. Prior to these works, a topological phase was synonymous with a non-trivial bulk-boundary correspondence. However, these two works put forward a broader definition, which says that a band insulator is in a topological phase if it cannot be continuously and adiabatically deformed to its atomic limit. This concept evolved in what is today known as topological quantum chemistry \cite{PoNatComm2017,BradlynNature2017,CanoPRL2018,VergnioryPRE2017}, which perhaps can be defined as the science of identifying topological bands in stoichiometric condensed substances. It comes with fine tools that have been combined with first-principle computer simulations of quantum solids to assess the topology of the energy bands in large classes of stoichiometric materials \cite{WiederScience2018,VergnioryNature2019,ElcoroNatComm2021,
WiederNatRevMat2022,VergnioryScience2022}. The topological criterium in these works is the pure homotopy equivalence of the bands, while ours is stable homotopy. The latter allows new degrees of freedom to participate in the deformation processes of the models and, as a result, our predictions hold even when the physical systems interact with external structures. Furthermore, we do not identify atomic limits or place labels like topological and trivial and, instead, we simply calculate the equivalence classes w.r.t. the stable homotopy of the spectral projections generated from the algebra of dynamical matrices. K-theory then supplies the complete invariants of these classes, hence enabling us to see if two spectral projections can or can not be stably deformed into each other. Lastly, we are seeking to implement a bottom-up approach, where we design meta-materials that deliver the topological bands, as opposed to the top-bottom approach in \cite{WiederScience2018,VergnioryNature2019,ElcoroNatComm2021,
WiederNatRevMat2022,VergnioryScience2022}, where large libraries of existing materials were scrutinized for topological bands. Of course, the latter still represent an extremely valuable source of information for meta-material scientists.

\vspace{0.1cm}

We now turn our attention to the mathematical physics literature, specifically to the works by Shiozaki et al \cite{ShiozakiPRB2014,ShiozakiPRB2016,ShiozakiPRB2017}, which initiated the systematic applications of topological K-theoretic methods to systems with space group symmetry. These works include the enrichment by the fundamental symmetries and treat all Wyckoff positions at once. It is shown there that, after a Bloch decomposition, the spectral bands are classified by twisted equivariant K-theories (as introduced in \cite{FreedAHP2013}; see also \cite{GomiSIGMA2017}) over the Brillouin torus w.r.t. to actions of the point group. These K-theories are carefully formulated in generic setting and \cite{ShiozakiPRB2017} explicitly computes them for all 17 wallpaper groups in class A and class AIII. The results revealed an extremely rich landscape of topologically distinct phases, which can be a great source of inspiration not only for electronic materials but also for metamaterials where dynamics is carried by classical degrees of freedom (see also \cite{GomiLMP2019,GomiJGP2019}). Certainly, our setting is much more restricted, we deal with frieze groups instead of wallpaper groups and even then only with specific patterns generated as described below, not general systems symmetric under the action of a frieze group. The major difference, however, is that we use operator K-theory and this approach enables us to develop a real-space picture, where the connection between the K-theoretic predictions and their physical realizations is quite transparent. For example, we produce explicit projections from the algebra of dynamical matrices, whose regular representations supply concrete physical models for topological phase predicted by the K-theory (see section~\ref{Sec:Numerical}). Furthermore, our numerical algorithm for computing the K-theoretic labels of a generic spectral band avoids any Bloch decomposition and is entirely developed in the real-space picture (see section~\ref{Sec:NumInv}). This is important for us because this real-space picture can be further developed to include disorder, on which we will report in the near future.

\vspace{0.1cm}

The pure mathematics literature abounds with results on the K-theories of group $C^\ast$-algebras. One of the most effective tools of computations is the Baum-Connes assembly map \cite{BaumIHES1982,BaumProceedings1988,Baum1987,
BaumContMath1994,BaumEM2000}, which more often than not reduces the task to computing equivariant K-homologies of topological spaces (see \cite{AparicioANC2019} for a status report). Davis and L\"uck \cite{DavisLueck} have unified several existing assembly maps, which then enabled L\"uck and Stamm \cite{LueckStamm} to formally derive the K-theories of all crystallographic groups in arbitrary dimensions and, explicitly, the K-theories of all wallpaper groups (see also \cite{YangThesis}). While we hope that these results and methods will soon become a major source of inspiration in materials science, we will use them here only for validations and comparisons. This is because we want to showcase for the reader the simpler but less powerful computations based on the equivariant Chern character \cite{Baum1987,Lueck2002} (see Th. 6.1 of \cite{Mislin03}), which can only resolve the non-torsion component of the $K_0$-groups.\footnote{This is entirely sufficient in our case.} 

\vspace{0.2cm}

Resolving just the $K_0$-groups is not sufficient because we also need to identify explicit bases, represented by explicit elements from the stabilized algebra. As already pointed out in \cite{Echterhoff2010}, the Baum-Connes machinery is not always an effective tools in this respect. For crystallographic groups of low dimensions, the methods based on non-commutative CW-complexes developed in \cite{McAlisterPhD2005} are capable to deliver both the $K_0$-groups and sets of generators. Other more direct methods for specific crystallographic groups can be found in \cite{Cuntz82,HadfieldJOP2004}. We will use some of these sources when listing the generator of the $K_0$-groups of the frieze group $C^\ast$-algebras. While this is sufficient for the simple frieze groups invoked by our study, more systematic methods for resolving the bases of the $K_0$-groups are yet to be developed for the space groups in higher dimensions.

\section{Symmetric patterns of resonators on a strip}
\label{Sec:SymmetricPatterns}

A strip is a quasi one-dimensional slice of the Euclidean plane, such as ${\mathcal S} := {\mathbb R} \times [-1,1]$, equipped with the inherited Euclidean metric. 
Its complete group of isometries is generated by continuous translations along the long axis and reflections relative to the horizontal and vertical axes. This continuous group of isometries accepts seven distinct discrete subgroups, the frieze groups \cite{ConwayBook}. 
As we already announced in our introduction, we want to investigate the collective dynamics of harmonic oscillators arranged in patterns that are symmetric against the strip transformations contained in a frieze group. This section supplies a brisk review of the frieze groups and introduces our specific algorithm that creates the patterns. The section also analyzes the dynamical matrices of collections of harmonic resonators arranged in such symmetric patterns.

\subsection{Discrete symmetries of a strip}\label{Sec:frieze} The seven discrete subgroups of the full group of isometries of the strip are all listed in table~\ref{Ta:1}, together with their isomorphism classes, sets of generators, and actions on the strip. 
In this table, $\tau$ is the translation by one unit along the strip, while $\sigma_h$ and $\sigma_v$ are the reflections against the horizontal and vertical axes of the strip, respectively.

\vspace{0.1cm}

{\small
\begin{table}
  \caption{\small List of the frieze groups, their isomorphism classes, generators and actions on the plane.}
  \centering
  \begin{tabular}{ccclccc}
  \toprule
  Iso. class & \ \ \ \  $\Gamma$ \ \ \ \ & \ \ \ \ Generators \ \ \ \ & Orbifold    \\ \midrule  \addlinespace[0.5em]
  $\mathrm{Z}_\infty$  & $p1$ &   $\tau$ & \fgfx{f_p1}     \\  \addlinespace[0.5em]
  & $p11g$ &  $\tau_{1/2} \sigma_h$ & \fgfx{f_p11g}  \\ \addlinespace[0.5em] \midrule
 $\mathrm{Dih}_\infty$  & $p1m1$ & $\tau,\sigma_v$ & \fgfx{f_p1m1}    \\ \addlinespace[0.5em]
  & $p2$ & $\tau,\sigma_h \sigma_v$ & \fgfx{f_p2}   \\ \addlinespace[0.5em]
  & $p2mg$ & $\tau_{1/2}\sigma_h, \sigma_v$ & \fgfx{f_p2mg}  \\ \midrule \addlinespace[0.5em]
  $\mathrm{Z}_\infty \times \mathrm{Dih}_1$  & $p11m$ & $\tau, \sigma_h$ & \fgfx{f_p11m}   \\ \midrule \addlinespace[0.5em]
  $\mathrm{Dih}_\infty \times \mathrm{Dih}_1$   & $p2mm$ & $\tau, \sigma_h, \sigma_v$ & \fgfx{f_p2mm}   \\ \midrule \addlinespace[0.5em]
  \end{tabular}
  \label{Ta:1}
\end{table}
}

The structure of the frieze groups can be summarized in a concise form by presenting each of them in terms of their generators and relations:
\begin{enumerate}
\item $ p1  = \langle u \rangle,  \ u = \tau;$
\item $p11g  = \langle u \rangle,  \ u = \tau_\frac{1}{2} \sigma_v;$ 
\item $p1m1  = \langle u,v_1 ~|~ v_1^2, \ (u v_1)^2\rangle, \ u = \tau, \ v_1 =\sigma_v;$
\item $p2 = \langle u,v_1 ~|~ v_1^2, \ (u v_1)^2\rangle, \ u = \tau, \ v_1=\sigma_h\sigma_v$;
\item $p2mg = \langle u,v_1 ~|~ v_1^2, \ (u v_1)^2\rangle, \ u = \tau_{1/2} \sigma_h, \  v_1 = \sigma_v$;
 \item $p11m  = \langle u,v_2 ~|~ v_2^2, \ u v_2 u^{-1} v_2 \rangle, \ u = \tau, \ v_2 =\sigma_h;$
 \item $p2mm  = \langle u,v_1,v_2 ~|~ v_j^2, (v_1 v_2)^2, (u v_1)^2, u v_2 u^{-1} v_2\rangle,  \ u = \tau, \ v_1 = \sigma_v, \  v_2 = \sigma_h.$
\end{enumerate}
Thus, the discrete subgroups fall into four isomorphism classes, as already indicated in Table~\ref{Ta:1}. Furthermore, we can use a uniform notation to specify an element of any of the frieze groups as $u^n v_1^{\alpha_1} v_2^{\alpha_2} : = u^n v^\alpha$, where it is understood that, for example for $p1m1$ group, $\alpha_2$ takes only value 0. Here $\alpha=(\alpha_1,\alpha_2)$ and $v^\alpha = v_1^{\alpha_1} v_2^{\alpha_2}$. In this presentation, the composition rule is simply
\begin{equation}\label{Eq:Compo1}
\big (u^n v^\alpha\big) \cdot \big ( u^m v^\beta \big) = u^{n+(-1)^{\alpha_1} m} \; v^{(\alpha + \beta){\rm mod}2},
\end{equation}
and inversion
\begin{equation}\label{Eq:Inversion}
(u^n v^\alpha)^{-1} = u^{(-1)^{\alpha_1}} v^\alpha.
\end{equation}
The above rules supply a practical way to encode the tables of the frieze groups, which will be employed by our numerical simulations as explained in section~\ref{Sec:Numerical}.

\subsection{The class of our symmetric patterns}\label{Sec:Gen} Our algorithm for creating patterns that are symmetric to a particular frieze group $F$ consists of the following steps:
\begin{itemize}
\item Start from a finite number $N$ of seed shapes, placed at desired non-over- lapping locations and with desired orientations;
\item Apply the plane transformations contained in a frieze group to each of the seeds.
\item Adjust the seeds if the shapes of the resulting pattern overlap and repeat the process.
\end{itemize}

\begin{figure}
\center
  \includegraphics[width=0.99\linewidth]{Patterns.pdf}
  \caption{\small Examples of patterns generated from three seed triangles.}
 \label{Fig:1}
\end{figure}

There are several observations to be made about {\it our} symmetric patterns:
\begin{enumerate}[a)] 
\item As stated in our last rule, we will be avoiding those particular seed configurations that lead to overlapping shapes when acted with the frieze transformations. In the language of crystallography, we are therefore placing the seeds at so-called general Wyckoff positions that are fixed only by the action of the identity operation of the frieze group.
\item Given our choice, there are symmetric patterns that are not included in {\it our} class of symmetric patterns. Those are the patterns where a resonator sits at a higher symmetry point and the resonator has a shape that is symmetric relative to the particular site group symmetry. Note that these patterns cannot be continuously deformed into patterns from our class, because once a resonator is moved away from a higher symmetry point, symmetry requires that the number of resonators be immediately increased by a factor equal to the order of the site symmetry group. Therefore, the patterns with resonators placed at higher symmetry points live in their own separate worlds and are not investigated in this work. They are certainly very interesting and they deserve a full separate investigation.
\item Any of {\it our} patterns is completely determined by the shape, position, orientation of the seeding shapes. Hence, these three factors determine the configuration space of our patterns and any continuous deformation of a pattern can be thought as a curve in this configuration space. In the next section, we will add the internal structure of the resonators to this configuration space. This together with the constraints mentioned above completely specify the type of deformations we allow on the metamaterials.

\end{enumerate}

Examples of symmetric patterns generated from three seed triangles are shown in Fig.~\ref{Fig:1}. 



\subsection{Symmetric patterns of resonators}\label{Sec:SymmPatt} We now ask the reader to imagine that the shapes, like the one seen in Fig.~\ref{Fig:1}, are in fact physical resonators. 
For example, they can be steel plates embedded in an elastic medium, acoustic cavities, or magnetized solid bodies (see section~\ref{Sec:PlateResonators} for concrete examples). 



By construction, each resonator of the pattern can be conveniently and uniquely labeled by $(i,f) \in \{1,\ldots,N_s\} \times F$, where $i$ corresponds to the seed resonator on which the transformation $f$ was applied. We emphasize that this is a consequence of our decision to exclude any overlap of the resonators and, certainly, the above labeling breaks down if a seed resonator is placed at a point with non-trivial site symmetry group. 

\vspace{0.1cm}

Each resonator carries a finite number $K$ of relevant internal resonant modes. These modes will always be observed and measured from a frame rigidly attached to the resonator. This enables us to choose the bases for the internal spaces of resonators in a coherent fashion. Specifically, for the seed resonator $i$, we choose once and for all a basis for its space of internal resonant modes, which we denote by $|\alpha,i,f=1\rangle$, $\alpha =\overline{1,K}$. When acting with a plane transformation $f$ from the frieze group, we produce a resonator placed at proper position and properly rotated. When the resonances of this resonators are observed from the frame rigidly attached to it, they will be identical to that of the seed resonator. Thus, we can choose a natural basis for the space of internal resonant modes of $(i,f)$ resonator that is identical to $|\alpha,i,f=1\rangle$, $\alpha =\overline{1,K}$, when observed from the rigidly attached frame. We denote this natural basis by $|\alpha,i,f\rangle$, $\alpha =\overline{1,K}$. Note that this coherent labeling of the modes always exist for our special class of patterns but it does not exist for patterns with resonators sitting at non-generic Wyckoff positions. 

\begin{remark}{\rm For atomic systems, the orbitals are always observed and quantified in the laboratory frame, hence quite differently from what we are proposing here for classical resonators.
}$\Diamond$
\end{remark} 

The collective dynamics of a pattern of resonators takes place inside the Hilbert space ${\mathbb C}^N \otimes \ell^2(F)$, $N = N_s \, K$, spanned by the vectors $\xi \otimes |f\rangle$ with $\xi \in {\rm Span}\{|\alpha,i\rangle, \ \alpha=\overline{1,K}, \ i = \overline{1,N_s} \} \simeq {\mathbb C}^N$ and $f \in F$. Given our convention, a plane transformation acts as 
\begin{equation}\label{Eq:PlaneTf}
T_{f'} (\xi \otimes |f\rangle) = \xi \otimes |f' f \rangle .
\end{equation}
It is important to note that the vector $\xi$ is not acted on by the transformation. Now, like any bounded operator over ${\mathbb C}^N \otimes \ell^2(F)$, a dynamical matrix assumes the generic form
\begin{equation}\label{Eq:GenD}
H = \sum_{f,f' \in F} w_{f,f'} \otimes |f \rangle \langle f' |, \quad w_{f,f'} \in M_{N}({\mathbb C}),
\end{equation}
where $M_{N}({\mathbb C})$ denotes the algebra of $N \times N$ matrices with complex entries. As an experimental fact, the coupling matrix $w_{f,f'}$ becomes weaker as the distance between a pair $(f,f')$ of resonators is increased and in fact the coupling matrix cannot be experimentally resolved beyond a certain separation distance. Thus, for typical experiments, it is justified to assume that there is a finite number of terms in Eq.~\ref{Eq:GenD} involving one $f \in F$.

\vspace{0.1cm}

If the physical processes involved in the coupling of resonators are Galilean invariant, then the coupling matrices must satisfy the following symmetry constraints
\begin{equation}\label{Eq:Invariance}
w_{g \cdot f, g \cdot f'} = w_{f,f'}, \quad \forall \ f,f',g \in F.
\end{equation}
It is quite important to understand that this is not an assumption but rather a physical reality (at least for mechanical systems).  Indeed, let  there be some underlying Galilean invariant physical laws ${\mathcal P} \mapsto w({\mathcal P})$ which associate to each admissible pattern ${\mathcal P}$ of resonators a set of coupling matrices. Then, if $g$ is any Euclidean transformation, we must have
\begin{equation}
w_{g \cdot f, g \cdot f'}(g \cdot {\mathcal P}) = w_{f,f'}({\mathcal P}).
\end{equation}
Indeed, any instrumentation rigidly attached to the pair $(f,f')$ of resonators will not be able to sense the Euclidean drift $g$ of the entire platform. Now, if $g$ belongs to the frieze group that produced the pattern in the first place, then $g \cdot {\mathcal P} = {\mathcal P}$ and the Eq.~\eqref{Eq:Invariance} follows.

\begin{remark}{\rm As opposed to atomic systems, no additional conjugation with a representation of $g$ on the space ${\mathbb C}^N$ is needed in Eq.~\eqref{Eq:Invariance}, because the pair of resonators and their resonant modes are rigidly rotated by $g$ and the modes are observed and quantified from frames rigidly attached to the resonators. Another way to justify the absence of such conjugation is via the specific action~\eqref{Eq:PlaneTf} of the plane transformations on our basis.
}$\Diamond$
\end{remark} 

As a direct consequence of Eq.~\eqref{Eq:Invariance}, any dynamical matrix over one of our symmetric patterns can be always reduced to the following particular form
\begin{equation}\label{Eq:ReducedH}
H = \sum_{f,f' \in F} w_{1,f^{-1} \cdot f'} \otimes |f \rangle \langle f' |.
\end{equation}
This is already a strong indication that the dynamical matrices of our patterns of resonators form a sub-algebra of the bounded operators over ${\mathbb C}^N \otimes \ell^2(F)$. Computing this sub-algebra is the subject of the next section.

\begin{remark}{\rm The algebra ${\mathbb B}\big (\ell^2(F)\big )$ of bounded operators over $\ell^2(F)$ is very large and it has a trivial K-theory. Thus, in order to predict topological phenomena in the dynamics of the resonators, it is paramount to demonstrate that the algebra of dynamical matrices is actually a separable $C^\ast$-subalgebra of ${\mathbb B}\big (\ell^2(F)\big )$.
}$\Diamond$
\end{remark}


\section{Group \texorpdfstring{$C^\ast$-algebras}{C*-algebras} as natural environments for our symmetric models}
\label{Sec:GCAlg}

This section establishes the connection between the abstract $C^\ast$-algebra of the frieze groups and the algebra of dynamical matrices of the symmetric patterns of resonators introduced in the previous section.

\subsection{Group $C^\ast$-algebras: Facts and notations}\label{Sec:CG} There will be several groups involved in our analysis and, for this reason, we want to introduce some uniform notation that can be used exchangeably among the groups. The standard material of this subsection can be found, for example, in \cite{DavidsonBook}[Ch.~VIII].

\vspace{0.1cm}

Henceforth, given a generic discrete group $G$, its group algebra ${\mathbb C} G$ consists of formal series
\begin{equation}\label{Eq:CG}
q = \sum_{g \in G} \alpha_g \, g, \quad \alpha_g \in {\mathbb C},
\end{equation}
where all but a finite number of terms are zero. 
The addition and multiplication of such formal series are defined in the obvious way, using the group and algebraic structures of $G$ and ${\mathbb C}$, respectively. 
In addition, there exists a natural $\ast$-operation
\begin{equation}
q^\ast = \sum_{g \in G} \alpha_g^\ast \, g^{-1}, \quad (q^\ast)^\ast = q, \quad (\alpha q)^\ast = \alpha^\ast q^\ast, \ \alpha \in {\mathbb C}.
\end{equation}
Hence, ${\mathbb C} G$ is a $\ast$-algebra in a natural way.

\vspace{0.1cm}

We denote by $e$ the neutral element of $G$. Then the map
\begin{equation}
{\mathcal T} : {\mathbb C} G \to {\mathbb C}, \quad {\mathcal T}(q) = \alpha_e
\end{equation}
defines a faithful positive trace on ${\mathbb C} G$ and a pre-Hilbert structure on ${\mathbb C} G$ via
\begin{equation}
\langle q, q' \rangle : = {\mathcal T}(q^\ast q'), \quad q,q' \in {\mathbb C} G.
\end{equation}
The completion of the linear space ${\mathbb C} G$ under this pre-Hilbert structure supplies the Hilbert space $\ell^2(G)$, spanned by $|g\rangle$, $g \in G$, which form an orthonormal basis
\begin{equation}
\langle g , g' \rangle = \delta_{g,g'}, \quad g,g' \in G.
\end{equation} 
The left action of ${\mathbb C} G$ on itself can be extended to the action of a bounded operator on $\ell^2(G)$, and this supplies the left regular representation $\pi_L$ of ${\mathbb C} G$ inside the algebra ${\mathbb B}\big(\ell^2(G)\big)$ of bounded operators. Specifically,
\begin{equation}\label{Eq:PiLeftReg}
\pi_L(q) |g\rangle = \sum_{g'\in G} \alpha_{g'} |g' g \rangle = \sum_{g'\in G} \alpha_{g'g^{-1}} |g' \rangle, \quad q \in {\mathbb C} G, \quad g \in G,
\end{equation} 
where $q = \sum_{g' \in G} \alpha_{g'}  g'$ with $\alpha_{g'} \in {\mathbb C}$.
The completion of ${\mathbb C} G$ with respect to the norm
\begin{equation}
\|q\| : = \|\pi_L(q)\|_{{\mathbb B}(\ell^2(G))}
\end{equation}
supplies the reduced group $C^\ast$-algebra $C^\ast_r(G)$ of $G$.

\begin{remark}{\rm For any state on a $\ast$-algebra, there are left and right GNS representations. The right GNS representation $\pi_R$ corresponding to ${\mathcal T}$ has the same Hilbert representation space, but the elements act from the right (see Eq.~\eqref{Eq:PiRightReg}).
}$\Diamond$
\end{remark}

\begin{remark}\label{Re:TT}{\rm The following detail is worth mentioning. If the group $G$ is finite, then
\begin{equation}
{\mathcal T}(q) = \frac{1}{|G|} {\rm Tr}\big ( \pi_L(q) \big ) = \frac{1}{|G|} {\rm Tr}\big ( \pi_R(q) \big ),
\end{equation}
which can be seen straight from \eqref{Eq:PiLeftReg}.
}$\Diamond$
\end{remark}

Since we only deal here with discrete amenable groups, the reduced and full group $C^\ast$-algebras coincide. This is important because the relation between the group and its $C^\ast$-algebra is functorial only for the latter, in general. So, in the present context, we have the following simple but essential statement:

\begin{proposition}[\cite{NassopoulosIJCMS2008}]\label{Pro:LiftMor} Let $G$ and $H$ be discrete amenable groups and $\rho : G \to H$ be a group morphism. Then $\rho$ lifts to a $C^\ast$-algebra morphism $C^\ast_r(G) \to C^\ast_r(H)$, extending the map between group algebras
\begin{equation}
{\mathbb C} G \ni \sum_{g \in G} \alpha_g \, g \mapsto \sum_{g \in G} \alpha_g \, \rho(g) \in {\mathbb C} H.
\end{equation}
\end{proposition}

\subsection{Symmetric models} As we have seen, the group $G$ acts on itself via an action either from the left or from the right. The left action was already used to define the canonical left regular representation $\pi_L$ of the group $C^\ast$-algebra. The symmetry operations associated to the group elements are implemented via the left-regular representation:
\begin{equation}
U : G \to {\mathbb B}\big ( \ell^2(G) \big ), \quad U_g |g' \rangle : = \pi_L(g)|g'\rangle = |gg'\rangle.
\end{equation}
The right regular representation of the group algebra acts as
\begin{equation}\label{Eq:PiRightReg}
\pi_R(q) |g'\rangle : = \sum_{g\in G} \alpha_{g} |g' g^{-1} \rangle = \sum_{g\in G} \alpha_{g^{-1} g'} |g \rangle, \quad q \in {\mathbb C} G, \quad g' \in G.
\end{equation} 
Since the left and right group actions commute, we automatically have that
\begin{equation}
U_g^\ast \, \pi_R(q) \, U_g = \pi_R(q), \quad q \in C^\ast_r(G), \quad g \in U_g.
\end{equation}
Hence, the group $C^\ast$-algebra supplies models that are naturally symmetric w.r.t. the group $G$, via a mechanism that rests entirely on the associative property of the group algebra. 

\begin{remark}{\rm The following simple observation establishes a useful link between projections and representations. Suppose $p$ is a projection from $C^\ast_r(G)$. Then $\pi_R(p)\ell^2(G)$ is an invariant subspace for the group symmetries
\begin{equation}
U_g \big [ \pi_R(p)\ell^2(G) \big ] = \pi_R(p)\ell^2(G), \quad \forall \ g \in G.
\end{equation}
As such, $g \mapsto  \pi_L(g) \pi_R(p)$ supplies a representation of $G$ on the subspace $\pi_R(p)\ell^2(G)$. This simple observation will be exploited here quite often.
}$\Diamond$
\end{remark}


The above observations do not imply that the dynamical matrices commuting with the symmetries $U_g$ are necessarily from $C^\ast_r(G)$. For this, we need to take a closer look at the  right regular representations
\begin{equation}
\pi_R\Big (\sum_g \alpha_g \, g\Big ) = \sum_{g,g' \in G} \alpha_{g^{-1}g'} |g\rangle \langle g'|.
\end{equation}
They can be naturally extended to the algebra $M_N({\mathbb C}) \otimes C^\ast_r(G)$ over the Hilbert space ${\mathbb C}^N \otimes \ell^2(G)$. An element from $M_N({\mathbb C}) \otimes C^\ast_r(G)$ takes the same form as displayed in Eq.~\ref{Eq:CG}, but with coefficients $\alpha_g$ from $M_N({\mathbb C})$. Then the extension of the right regular representation becomes 
\begin{equation}\label{Eq:RRRep}
\pi_R\Big (\sum_g \alpha_g \, g\Big ) = \sum_{g,g' \in G} \alpha_{g^{-1}g'} \otimes |g\rangle \langle g'|.
\end{equation}
Similarly, the group symmetries $U_g$ can be lifted as $I \otimes U_g$ on the Hilbert space ${\mathbb C}^N \otimes \ell^2(G)$ by letting $G$ act trivially on ${\mathbb C}^N$.

\vspace{0.1cm}

Now, comparing the generic form Eq.~\eqref{Eq:ReducedH} of the dynamical matrices for Galilean invariant theories with the right regular representation~\eqref{Eq:RRRep}, we see that the two coincide if we take $G=F$ and $w_{f,f'} = \alpha_{f^{-1}f'}$. The important conclusion is that {\it all} dynamical matrices of the patterns of resonators proposed by us can be generated from the right regular representation of $C^\ast_r(F)$. Furthermore, we can be sure that our allowed deformations of the physical systems, which involve changing the shape, position and orientation of the seeding shapes, as well as the internal structure of the resonators, take place inside the stabilization of the $C^\ast$-algebra $C^\ast_r(F)$.

\begin{remark}{\rm Throughout, we will reserve the symbols $\pi_R$, ${\mathcal T}$ and $U$ for the objects introduced in this section, making sure that the group $G$ can be clearly read off from the context. In fact, since the left regular representation is only used in tandem with $U$, we simplify the notation for the right regular representation to $\pi$.
}$\Diamond$
\end{remark}

Specializing now for frieze groups, the elements of $M_{N}({\mathbb C}) \otimes C^\ast_r(F)$ can be presented as convergent infinite series
\begin{equation}\label{Eq:Elem1}
\sum_{n \in {\mathbb Z}} \sum_{\alpha \in \{0,1\}^2} w(n,\alpha) u^n v^{\alpha},
\end{equation}
with coefficients $w(n,\alpha)$ from $M_{N}({\mathbb C})$. Given the rule~\eqref{Eq:PiRightReg}, the right regular representation of an element $h$ as in Eq.~\eqref{Eq:Elem1} has matrix elements
\begin{equation}\label{Eq:MatElem}
\langle i,  \big ( u^m v^\beta \big) \cdot \big (u^n v^\alpha \big)^{-1} | \pi(h) |j, u^m v^\beta \rangle = w_{ij}(n,\alpha).
\end{equation}
By sampling $i$, $j$, $(n,\alpha)$ and $(m,\beta)$ over all allowed values, Eq.~\eqref{Eq:MatElem} returns all the matrix elements of the element $h$.

\vspace{0.1cm}

As seen in the above paragraph, the group itself supplies very natural labels for the resonators of a symmetric pattern and the group algebra provides a very efficient way to store or to generate dynamical matrices. Furthermore, with the natural labels, the matrix elements can be read off from the coefficients of the series~\eqref{Eq:Elem1} and from the multiplication table of the group. We recommend the experimental scientists planning to build laboratory models to fully embrace this practice. After all, we are just recommending that, instead of labeling the resonators by $1,2,etc$ as it is typically done, to label the resonators by the group element that placed a resonator at its current position, by acting on the corresponding seed. 

\begin{remark}{\rm At this stage, it should become quite clear that all the above break down if a seed resonator is placed at a point with non-trivial site group.
}$\Diamond$
\end{remark} 


\section{Topological dynamics by \texorpdfstring{$K$-theory}{K-theory}} 
\label{Sec:TopologicalDynamics}

\subsection{Why $K$-theory?}\label{Sec:Why} Suppose we are dealing with a single symmetric model $h$ for which we already computed the resonant spectrum ${\rm Spec}\big(\pi(h)\big )$. 
Furthermore, suppose this spectrum displays  a number of spectral bands separated by spectral gaps, which will be referred below as gapped spectral bands. 
As it is well known, if we excite the system in a state belonging to the subspace $P {\mathcal H}$, with ${\mathcal H}={\mathbb C}^N \otimes \ell^2(F)$ the Hilbert space of the model and $P$ the spectral projection onto a band, then the time-evolved state cannot escape the space $P {\mathcal H}$. Thus, we can think of $P {\mathcal H}$ and of the corresponding spectral band as topological invariants for the dynamics. We can also focus on a spectral gap and on its gap projection, {\it i.e.} the spectral projection onto the spectrum below that gap, for which the conclusions will be the same. We, however, choose here to focus on the gapped bands and their projections. Suppose now that we are dealing with two models, $h_1$ and $h_2$, each displaying two gappped spectral bands with corresponding spectral projections $P_1$ and $P_2$. We can ask the question if we can continuously deform one model into another while keeping the spectral bands gapped. If such action is possible, then the adiabatic theorem \cite{AvronCMP1987} assures us of the existence of a unitary operator $U$, the adiabatic time evolution, such that $U(P_1 {\mathcal H})=P_2 {\mathcal H}$. Thus, in such situations, we can map any dynamics excited from $P_1 {\mathcal H}$ into a dynamics excited from $P_2 {\mathcal H}$ and we can declare that the dynamical features that can be observed in the two models, when excited from these gapped bands, are qualitatively the same.

\vspace{0.1cm}

K-theory is a computational tool designed to answer the above question. Indeed, given two band projections acting on ${\mathbb C}^N \otimes \ell^2(F)$ and ${\mathbb C}^{N'} \otimes \ell^2(F)$, hence coming from symmetric models with possibly different degrees of freedom, K-theory declares the two projections to be equivalent if we can continuously deform one into the other, with the deformations taking place over any larger Hilbert space ${\mathbb C}^{M} \otimes \ell^2(F)$ that embeds the original Hilbert spaces. In other words, the models are deformed inside the larger algebra ${\mathbb K} \otimes C^\ast_r(F)$, with ${\mathbb K}$ begin the algebra of compact operators over $\ell^2({\mathbb N})$, which embeds all finite matrix algebras. 
Almost tautologically, the equivalence class $[P]_0$ of a projection, relative to the equivalence relation we just stated, defines the complete topological invariant associated to the projection. 
Thus, if the goal is to classify the dynamics by topological means, then one has to map all the equivalence classes supported by the algebra that generates the models. The good news is that the set of K-theoretic equivalence classes accepts a natural abelian semigroup structure. 
Its enveloping group defines the $K_0$-group of the algebra and, as we shall see, this group admits a small number of generators. 
By identifying these generators, we resolve completely the topological classification of possible dynamical features supported by our symmetric models.

\begin{remark}{\rm In this work we will not deal at all with the $K_1$-group of the algebra, which becomes relevant if the so called chiral symmetry is imposed on the dynamical matrices \cite{ProdanSpringer2016}.
}$\Diamond$
\end{remark}

\subsection{$K$-theories by Baum-Connes machinery}
\label{Sec:BC} 

Since the K-theoretic ideas are being rapidly adopted by the condensed matter and metamaterial communities, we believe that the current context offers a good opportunity to showcase how the Baum-Connes machinery works. The Baum-Connes assembly map $\mu$ makes the connection 
\begin{equation}\label{Eq:BC}
K_0^F(\underline{E}F) \stackrel{\mu}{\to} K_0\big(C_r^\ast(F)\big)
\end{equation}
between the equivariant $K$-homology of a universal space $\underline{E}F$ for proper actions of the frieze group $F$ \cite{BaumContMath1994} and the $K_0$-group of the group $C^\ast$-algebra of $F$. The map $\mu$ in Eq.~\eqref{Eq:BC} is a group isomorphism because the frieze groups are amenable and every countable amenable discrete group is a-T-menable, in which case the Baum-Connes conjecture is known to hold \cite{HigsonKasparov1997}. Our first task is to decide on a model for $\underline{E}F$. For this, we indicate that the full group of isometries of the strip ${\rm Iso}({\mathcal S})$ is a Lie group with a finite number of connected components because it is the semidirect product of the Lie group ${\mathbb R}$ of translations and the finite group ${\mathbb Z}_2 \times {\mathbb Z}_2$, generated by the two reflections against the vertical and horizontal axes of the strip. Then, according to \cite{BaumContMath1994}[p.~8], the space of the left cosets 
\begin{equation}
{\rm Iso}({\mathcal S})/{\mathbb Z}_2 \times {\mathbb Z}_2 : = \{s \cdot {\mathbb Z}_2 \times {\mathbb Z}_2, \ s \in {\rm Iso}({\mathcal S})\}  \simeq {\mathbb R}
\end{equation} 
can serve as a universal space for proper actions of ${\rm Iso}({\mathcal S})$ and also for any of its discrete subgroups. Therefore, $\underline{E}F$ can be chosen as ${\mathbb R}$, with the action of $F$ induced from its left action on ${\rm Iso}({\mathcal S})/{\mathbb Z}_2 \times {\mathbb Z}_2$. Specifically, $\tau$ and $\sigma_v$ act as expected while $\sigma_h$ acts as the identity map.

\vspace{0.1cm}

Our next task is to compute the abelian groups $K_0^F(\underline{E}F)$ and the main tool here is the equivariant Chern character \cite{Baum1987,Lueck2002}, which supplies the isomorphism described in the following statement:

\begin{theorem}[Th.~6.1, \cite{Mislin03}] Let $X$ be a proper $F$-CW complex. Then
\begin{equation}\label{Eq:Chern}
  K_0^F(X) \otimes {\mathbb Q} \cong \bigoplus_{r\in {\mathbb N}} \bigoplus_{\gamma \in S F} H_{2r}(X^{\gamma}/Z(\gamma); {\mathbb Q}),
\end{equation}
where $S F$ is a set of representative members: one for each conjugacy class for each finite order element in $F$. 
$X^\gamma$ is the space of fix points of $X$ under the action of $\gamma$ and $Z(\gamma)$ is the centralizer of $\gamma$. The groups appearing on the right are the ordinary homology groups of spaces with constant coefficients in ${\mathbb Q}$.
\end{theorem}

\begin{remark}{\rm As already stressed in our introduction, the tensoring with ${\mathbb Q}$ washes out the torsion part of $K_0^F(X)$. However, for our concrete context, we know from other sources that the $K_0$-groups, hence also $K_0^F(\underline E F)$, are torsion free. In this case, Eq.~\eqref{Eq:Chern} holds without tensoring by ${\mathbb Q}$ and the K-homology groups can be taken with coefficients in ${\mathbb Z}$.
}$\Diamond$
\end{remark}

Based on the above observation and the fact that $\mu$ is an isomorphism in our specific context, we conclude
\begin{equation}\label{Eq:FinalChern}
  K_0\big (C^\ast_r(F)\big ) \cong \bigoplus_{r\in {\mathbb N}} \bigoplus_{\gamma \in S F} H_{2r}(\underline{E}F^{\gamma}/Z(\gamma)).
\end{equation}
As we shall see, the resulting fix point spaces $\underline{E}F^\gamma$ ($\underline{E}F={\mathbb R}$ as topological space) in our applications of the Baum-Connes machinery are homeomorphic either to ${\mathbb S}^1$ or to spaces contractible to a point, hence it is useful to recall that
\begin{equation}\label{Eq:HH}
  H_k({\mathbb S}^1) =
  \left\lbrace 
  \begin{array}{ll}
  {\mathbb Z}, & k =0; \\
  {\mathbb Z}, & k = 1; \\
  0,   & \text{otherwise}.
  \end{array}
  \right. \qquad
 H_k (\bullet ) =
 \left\lbrace 
 \begin{array}{ll}
 {\mathbb Z}, & k =0; \\
 0,   & \text{otherwise}.
 \end{array}
 \right.
\end{equation}


The computations of the right-hand side of Eq.~\eqref{Eq:Chern} are detailed below and the results are summarized in Table~\ref{TK0}. These computations are complemented with explicit lists of generators of the $K_0$-groups.

\begin{table}
  \caption{\small Summary of the $K$-theoretic calculations.}
  \centering
  \begin{tabular}{cccc}
    \toprule
  Iso. class & Abstract generators & $S F$ & $K_0(C^\ast_r F) $
  \\ \midrule
  $\mathrm{Z}_\infty$ & $u$  & $e$ & $\mathbb{Z}$ \\ 
  $\mathrm{Dih}_\infty$ & $u$, $v_1$ & $e, v_1, uv_1$ & $\mathbb{Z}^3$ \\ 
  $\mathrm{Z}_\infty \times \mathrm{Dih}_1$ & $u$, $v_2$ &  $e, v_2$ & $\mathbb{Z}^2$ \\ 
  $\mathrm{Dih}_\infty \times \mathrm{Dih}_1$ & $u$, $v_1$, $v_2$ & 
  $e, v_1, v_2, v_1v_2, uv_1, u v_1 v_2$ & $\mathbb{Z}^6$\\ 
  \end{tabular}
  \label{TK0}
\end{table}

\subsubsection{$\mathrm{Z}_\infty$ isomorphism class}

The groups $p1$ and $p11g$ are both isomorphic to the infinite cyclic group $\mathrm{Z}_\infty = \langle u \rangle \cong \mathbb{Z}$. There are no non-trivial finite order elements since $u^m = 1$ implies $m=0$. For both $p1$ and $p11g$, we have $\underline{E}F^{1}/Z(1)= \mathbb{R}/ \mathbb{Z} \cong {\mathbb S}^1$ and Eq.~\eqref{Eq:FinalChern} together with \eqref{Eq:HH} give
\begin{equation}
  K_0\big(C^\ast_r (p1) \big ) \cong K_0\big(C^\ast_r (p11g) \big ) \cong {\mathbb Z}.
\end{equation}
The generators of the $K_0$-groups can be represented by the identity element of the group $C^\ast$-algebras.


\subsubsection{$\mathrm{Dih}_\infty$ isomorphism class}

The groups $p1m1$, $p2$ and $p2mg$ are all isomorphic to the infinite dihedral group $\mathrm{Dih}_\infty$.
Their $K_0$-groups will thus be  isomorphic as well.
We recall the presentation
\begin{align}
  \mathrm{Dih}_\infty = \braket{ u, v_1 }{v_1^2 , (u v_1)^2},
\end{align}
and the fact that every group element can be written as $\gamma_{n,m} = u^n v_1^m$, where $n \in {\mathbb Z}$ and $m \in \lbrace 0,1\rbrace$.
Using the composition rule from Eq.~\ref{Eq:Compo1}, we obtain
\begin{align}
  \gamma_{n,m}^2 = u^n v_1^m u^n v_1^m
  = u^{n + (-1)^m n} v_1^{ 2 m~ {\rm mod}2}
  =
  \left\lbrace
  \begin{array}{ll}
    u^{2n}, &  m=0 \\
    e, & m=1
  \end{array}
  \right.
\end{align}
Therefore, $\gamma_{n,m}$ is of finite order whenever $m=0$ and $n=0$ or when $m=1$ for arbitrary $n$.
In those cases, we have $\gamma_{0,0} = e $ and $\gamma_n : =  \gamma_{n,1} = u^n v_1$. Furthermore,
\begin{align}
  \gamma_{p,q}^{-1} \gamma_n \gamma_{p,q}
  & = v_1^{-q} u^{-p}  u^n v_1 u^p v_1^q
  \notag \\
  & = v_1^{q}   u^{n-2p} v_1^{q+1}
  \notag \\
  & =    u^{(-1)^q (n-2p)} v_1 .
\end{align}
In particular, this means that we have the equivalence by conjugacy
\begin{equation}
  \gamma_n \sim \gamma_{n+2m} .
\end{equation}
Therefore, every finite order element is either conjugate to $1$, to $\gamma_0$ or to $\gamma_1$, and so $S(\mathrm{Dih}_\infty)= \lbrace e, \gamma_0=v_1, \gamma_1 = u v_1 \rbrace$.
For $q=0$, $\gamma_{p,q}$ is in the centralizer of $\gamma_n$ only if $p=0$, while for $q=1$,  $\gamma_{p,q}$ is in the centralizer of $\gamma_n$ only if $p=n$.
Therefore,
\begin{align}
  Z(\gamma_n) = \lbrace e, \gamma_n \rbrace \cong {\mathbb Z}_2 .
\end{align}
The fixed point spaces are $\underline{E}F^{e} = {\mathbb R}$, $\underline{E}F^{\gamma_0} = \{ 0 \}$, $\underline{E}F^{\gamma_1}= \{ 1/2 \}$, as for the quotient spaces we have $\underline{E}F^{ e}  / Z(e) = [0,1/2]\cong \lbrace\bullet\rbrace$ and obviously this is also true for the rest of them, $ \underline{E}F^{\gamma_0}  / Z(\gamma_0) \cong    \underline{E}F^{\gamma_1}  / Z(\gamma_1) \cong \lbrace\bullet\rbrace$. 
Lastly, Eqs.~\eqref{Eq:FinalChern} and \eqref{Eq:HH} give 
\begin{equation}
  K_0\big(C^\ast_r( \mathrm{Dih}_\infty)\big) 
  \cong  {\mathbb Z}^3.
\end{equation}

The computation of the above $K_0$-group and its generators appeared in \cite{Cuntz82} as an application  of the methods developed for free products, given that $C^\ast_r(\mathrm{Dih}_\infty))=C_r^\ast({\mathbb Z}_2) \star C_r^\ast ({\mathbb Z}_2)$. These generators have been also identified in \cite{HadfieldJOP2004} by more direct methods. They are:\begin{equation}
  p_1 = e, \ p_2 = \tfrac{1}{2}(e-v_1), \ p_3 = \tfrac{1}{2}(e- u v_1).
 \end{equation}

\subsubsection{$\mathrm{Z}_\infty \times \mathrm{Dih}_1$ isomorphism class}

Only the group $p11m$ belongs to this isomorphism class. We recall the presentation
\begin{align}
  \mathrm{Z}_\infty \times \mathrm{Dih}_1 = \braket{ u, v_2 }{v_2^2 ,  u v_2 u^{-1} v_2 }.
\end{align}
Every group element can be written as $\gamma_{n,m} = u^n v_2^m$, where $n \in {\mathbb Z}$ and $m \in \lbrace 0,1\rbrace$.
Via Eq.~\ref{Eq:Compo1}, we now obtain
\begin{align}
  \gamma_{n,m}^2 = u^n v_2^m u^n v_2^m
  =
 u^{2n} v_2^{2m}
 = u^{2n} .
\end{align}
Therefore, $\gamma_{n,m}$ is of finite order only if $n=0$, and we have
\begin{equation}
  S  (\mathrm{Z}_\infty \times \mathrm{Dih}_1)
  = \lbrace 
  e, v_2 \rbrace . 
\end{equation}
Furthermore, we find
\begin{align}
  \gamma^{-1}_{p,q}  v_2 \gamma_{p,q} 
  &=  v_2^{-q} u^{-p}  v_2 u^p v_2^q
  =  v_2^{-q}   v_2  v_2^q
  = v_2 ,
\end{align}
which implies that $Z(v_2) = \mathrm{Z}_\infty \times \mathrm{Dih}_1$.
Since $\underline{E}F^{e} =\underline{E}F^{v_2} = {\mathbb R}$, we obtain 
\begin{equation}
  \underline{E}F^e / Z(e)  =\underline{E}F^{v_2} / Z(v_2) =  {\mathbb S}^1.
\end{equation}
Again, employing Eqs.~\eqref{Eq:FinalChern} and \eqref{Eq:HH}, we find
\begin{equation}
  K_0\big(C^\ast_r (\mathrm{Z}_\infty \times \mathrm{Dih}_1)\big)  \cong  {\mathbb Z}^2.
\end{equation}
The generators of this $K_0$-group can be represented by
\begin{equation}
p_1 = e, \ p_2 = \tfrac{1}{2}(e-v_2).
\end{equation}

\subsubsection{$\mathrm{Dih}_\infty \times \mathrm{Dih}_1$ isomorphism class}

Only the group $p2mm$ belongs to this isomorphism class, and we have the presentation
\begin{equation}
  \mathrm{Dih}_\infty \times \mathrm{Dih}_1 =  
  \langle u,v_1,v_2 ~|~ v_j^2, (v_1 v_2)^2, (u v_1)^2, u v_2 u^{-1} v_2\rangle
\end{equation}
Every group element can be written in the form $\gamma_{k,l,m} = u^k v_1^l v_2^m$, where $k \in {\mathbb Z}$ and $l, m\in \lbrace 0, 1\rbrace $.
In this case, we find
\begin{align}
  \gamma_{k,l,m}^2 
  =
  \left\lbrace
  \begin{array}{ll}
    u^{2k}, &  l=0 \\
    e, & l=1.
  \end{array}
  \right.
\end{align}
Thus, $\gamma_{k,l,m}^2 = e$ if either (i) $l=1$ for arbitrary $k$ and $m$ or (ii) $l=0$ and $k=0$ for arbitrary $m$. First, we take a look at case (i) and determine the conjugacy classes and centralizers. Similar to before, we have
\begin{align}
  \gamma_{p,q,r}^{-1}
  \gamma_{k,1,m} \gamma_{p,q,r} 
 
 
 
 
 
 
 
 
 
 
  & = \gamma_{(-1)^q(k-2p),1,m}.
\end{align}
Therefore, we again obtain a conjugacy equivalence of the form
\begin{equation}
  \gamma_{k,1,m} \sim  \gamma_{\pm(k-2p),1,m} 
\end{equation}
We select $k=0$ and $k=1$ as representative elements and obtain the centralizers
\begin{equation}
Z(\gamma_{0,1,m}) = \lbrace
  1, v_1, v_2, v_1 v_2 \rbrace, \quad
  Z(\gamma_{1,1,m}) = \lbrace
  1, u v_1, v_2, u v_1 v_2 \rbrace.
\end{equation}
Both are isomorphic to the Klein 4-group.
The fix-point sets are
\begin{equation}
\underline{E}F^{\gamma_{0,1,0}} = \underline{E}F^{\gamma_{0,1,1}} = \{0\}, \quad \underline{E}F^{\gamma_{1,1,0}} = \underline{E}F^{\gamma_{1,1,1}} = \{1/2\}.
\end{equation}
In each case, the quotient space $\underline{E}F^\gamma / Z(\gamma)$ is contractible to a point.

\vspace{0.1cm}

Next, we turn to case (ii) and find
\begin{equation}
  \gamma_{p,q,r}^{-1}
  \gamma_{0,0,m}
  \gamma_{p,q,r}
  =
  \gamma_{0,0,m} ,
\end{equation}
and so $Z(\gamma_{0,0,m}) = \mathrm{Dih}_\infty \times \mathrm{Dih}_1 $ .
The fix-point set are
\begin{align}
  \underline{E}F^{\gamma_{0,0,0}} = \underline{E}F^{\gamma_{0,0,1}} = {\mathbb R},
\end{align}
such that
\begin{equation}
  \underline{E}F^{\gamma_{0,0,0}} / Z(\gamma_{0,0,0} ) = \underline{E}F^{\gamma_{0,0,1}} / Z(\gamma_{0,0,1} ) = [0, 1].
\end{equation}
Both quotient spaces are contractible to a point.

\vspace{0.1cm}

In summary, we have found
\begin{equation}
  S ( \mathrm{Dih}_\infty \times \mathrm{Dih}_1) = \lbrace  1, v_1, v_2, v_1 v_2, u v_1, u v_1 v_2 \rbrace 
\end{equation}
and all quotient spaces $\underline{E}F^{\gamma} / Z(\gamma )$ are contractible to a point for $ \gamma \in S ( \mathrm{Dih}_\infty \times \mathrm{Dih}_1) $. Then Eqs.~\eqref{Eq:FinalChern} and \eqref{Eq:HH} give
\begin{equation}
  K_0\big (C^\ast_r (\mathrm{Dih}_\infty \times \mathrm{Dih}_1)\big ) \cong  {\mathbb Z}^6.
\end{equation}

The six generators of the $K_0$-group can be chosen as
\begin{equation}\label{Eq:K0Generators}
  \begin{aligned}
  & p_1 = e, \ p_2 = \tfrac{1}{2}(e-v_1), \ p_3 = \tfrac{1}{2}(e- u v_1), \\
  & p_4 = \tfrac{1}{2}(e+v_2), \ p_5 = p_2(e- p_4), \ p_6 = p_3 (e-p_4).
  \end{aligned}
 \end{equation}
This can be seen by applying K\"unneth theorem  \cite{BlackadarBook}[Th.~23.1.2] to $C^\ast_r (\mathrm{Dih}_\infty \times \mathrm{Dih}_1) \simeq C^\ast_r (\mathrm{Dih}_\infty) \otimes C^\ast_r (\langle e,v_2\rangle)$.

\section{Algorithm for computing the K-theoretic labels}
\label{Sec:NumInv}

Given a band projection $\pi(p)$, its class in $K_0$-theory can be expressed in one of the bases $\{p_a\}$ listed in the previous section, $[p]_0 = \sum_a n_a [p_a]_0$, $n_a \in {\mathbb Z}$. From general considerations, we can be sure that two band projections fall into the same topological class if their K-theoretic labels $\{n_a\}$ all coincide. Our goal for this section is to supply practical algorithms for computing the K-theoretic labels of a band projection, generated by an arbitrary symmetric model. For this, we employ Kasparov's bivariant K-theory and in particular Kasparov's product\cite{KasparovJSM1981,KasparovJSM1987} for guidance and as a convenient book-keeping instrument, as well as to illustrate a model calculation that can be repeated for other group $C^\ast$-algebras. We use standard notation, {\it e.g.} as in \cite{BlackadarBook}, and a brief review of the concepts is supplied below.

\subsection{General considerations}\label{Sec:GeneralStr} We recall that, for ungraded $C^\ast$-algebras $A$ and $B$, the elements of Kasparov's abelian group $KK_0(A,B)$ can be presented as the homotopy class of a pair of morphisms $\bar \phi =(\phi_1,\phi_2)$ from $A$ to ${\rm End}^\ast\big (\ell^2({\mathbb N},B) \big )$ with the property that $\phi_1 -\phi_2$ takes values in ${\mathbb K} \otimes B$ \cite{CuntzKTheory1987}. In particular, $KK_0({\mathbb C},A)$ coincides with the K-group $K_0(A)$ and any morphism $\psi : A \to B$ supplies an element of $KK_0(A,B)$. If $A$, $B$, $C$ are $C^\ast$-algebras, there is an associative product
\begin{equation}
KK_0(A,B) \times KK_0(B,C) \to KK_0(A,C)
\end{equation}
with quite exceptional properties \cite{Cuntz2005}. In particular, if $\psi : A \to B$ and $\psi' : B \to C$ are $C^\ast$-algebra morphisms, then
\begin{equation}
[\psi ] \times [\psi'] = [\psi' \circ \psi].
\end{equation}
This particular case will actually cover all applications considered here. 

\vspace{0.1cm}

Now, let $F$ be any of the frieze groups. Our goal is to construct pairings of the type
\begin{equation}\label{Eq:Pairing1}
KK_0\big ({\mathbb C},C^\ast_r(F)\big ) \times KK_0\big (C^\ast_r(F), {\mathbb C}\big ) \to KK_0({\mathbb C},{\mathbb C}) \simeq {\mathbb Z},
\end{equation}
such that
\begin{equation}
[p_a]_0 \times [\bar \phi_b] = \Lambda \, \delta_{a,b}, \quad \Lambda \in {\mathbb N}^\times,
\end{equation}
where $\{[p_a]\}$ is one of the K-theoretic bases listed in the previous section. If $[p]_0 = \sum_a n_a [p_a]_0$, $n_a \in {\mathbb Z}$, is the decomposition of the $K_0$-class of a projection in such basis, then
\begin{equation}
\Lambda^{-1} \, [p]_0 \times [\bar \phi_a] = n_a
\end{equation}
and this will supply an algorithm to map the K-theoretic labels $\{n_a\}$, hence, all the topological invariants associated to a projection $p \in {\mathbb K} \otimes C_r^\ast(F)$.

\vspace{0.1cm}

Our strategy is to construct the pairings~\eqref{Eq:Pairing1} in two steps,
\begin{equation}\label{Eq:Pairing2}
KK\big ({\mathbb C},C^\ast_r(F)\big ) \times KK\big (C^\ast_r(F), C^\ast_r(\tilde F)\big ) \times KK\big (C^\ast_r(\tilde F), {\mathbb C}\big ) \to {\mathbb Z},
\end{equation}
where $\tilde F$ is a finite group. The pairings $KK\big ({\mathbb C},C^\ast_r(\tilde F)\big )\times KK\big (C^\ast_r(\tilde F), {\mathbb C}\big )$ are essentially known and computable once the representation theory of the finite group $\tilde F$ is resolved. Hence,  the stated strategy reduces to finding appropriate Kasparov cycles from $KK\big (C^\ast_r(F), C^\ast_r(\tilde F)\big )$. As we shall see, this cycle will take the form of a simple morphism. 

\subsection{Explicit pairings} As we have seen in section~\ref{Sec:SymmetricPatterns}, all frieze groups can be derived from generators and relations. In particular, they all have ${\mathbb Z}$, generated by the element $u$ in our notation, as a normal subgroup.

\begin{proposition} Let $F$ be any of the frieze groups and let $k{\mathbb Z}$ be the subgroup of $F$ generated by $u^k$ for some fixed $k \geq 2$.  Then $k {\mathbb Z}$ is a normal subgroup of $F$.
\end{proposition}

\begin{proof} We can treat all frieze groups at once by using the conventions stated in subsection~\ref{Sec:frieze}. Indeed, given the multiplication rule~\eqref{Eq:Compo1}, we have 
\begin{equation}
(u^n v^\alpha)^{-1} u^k (u^n v^\alpha) = v^\alpha u^k v^\alpha = u^{(-1)^{\alpha_1} k} \in k {\mathbb Z},
\end{equation}
which shows that, indeed, $k {\mathbb Z}$ is indeed a normal subgroup.
\end{proof}

\begin{corollary} There exist the group morphisms $F \to F/k {\mathbb Z}$ from the infinite frieze groups to finite groups. The latter correspond to adding the relation $u^k =1$ to the defining relations of the frieze groups. 
\end{corollary}

\begin{remark}{\rm The above statements will also be relevant for our numerical applications. Indeed, the quotient $F/k{\mathbb Z}$ supplies a finite group and a canonical morphism $F \to F/k {\mathbb Z}$, which deliver convenient finite size approximations for the infinite lattice models (see subsection~\ref{Sec:NA}).
}$\Diamond$
\end{remark}

According to Proposition~\ref{Pro:LiftMor}, these morphisms lift to morphisms $\varphi_k : C^\ast_r(F) \to C^\ast_r(F/k{\mathbb Z})$ of $C^\ast$-algebras, hence they supply Kasparov cycles $$[\varphi_k] \in KK\big ( C^\ast_r(F), C^\ast_r(F/k{\mathbb Z}) \big ).$$
For elements from $\ell^1(F)$, the morphism $\varphi_k$ can be written down explicitly as
\begin{equation}\label{Eq:PhiE}
\sum_{n\in {\mathbb Z}} \sum_{\alpha} c(n,\alpha) u^n v^\alpha  \mapsto \sum_{m=0}^{k-1} \sum_{\alpha} \Big (\sum_{r \in k {\mathbb Z}} c(m+r,\alpha) \Big ) \tilde u^m \tilde v^\alpha,
\end{equation}
where $\tilde u$ and $\tilde v_j$ are the classes of $u$ and $v_j$'s in $F/k{\mathbb Z}$. Below is an alternative characterization of the morphism $\varphi_k$:

\begin{proposition}\label{Pro:PhiKPiK} Let ${\mathcal T}_k$ be the canonical trace on ${\mathbb C} F/k {\mathbb Z}$. Then ${\mathcal T}_k \circ \varphi_k$ supplies a positive trace on ${\mathbb C} F$. Let $({\mathcal H}_k,\pi_k)$ be the right GNS representation of $C^\ast_r(F)$ for this tracial state. Then the image $\pi_k({\mathbb C} F)$ in ${\mathbb B}({\mathcal H}_k)$ coincides with the right regular representation of $C^\ast_r(F/k {\mathbb Z})$. Hence, $\varphi_k$ can be identified with the map $\pi_k$.
\end{proposition}

\begin{proof} First, let us point out that the trace ${\mathcal T}_k$ generates the right regular representation of $C^\ast_r(F/k{\mathbb Z})$ and, since $\varphi_k$ is surjective,   ${\mathcal H}_k$ coincide with $\ell^2(F/k {\mathbb Z})$, the Hilbert space of the right regular representation. Then a direct computation shows that $\pi_k(f) = \pi \big (\varphi_k(f)\big )$ for any $f \in {\mathbb C} F$, where $\pi$ is the right regular representation of $F/k{\mathbb Z}$.
\end{proof}

\iffalse
\begin{proposition} Let $F$ be any of the frieze groups and let $S$ be the subgroup of $F$ generated by the symmetries related to the  $K_0$-bases, as discussed in the previous section. Then $\tilde F : = F/2 {\mathbb Z}$ is canonically isomorphic to $S$.
\end{proposition}

\begin{proof} We consider first the ${\rm Dih}_\infty$ isomorphism class. We have
\begin{equation}
\tilde F = \langle \tilde u,\tilde v_1 \; | \; \tilde u^2,  \tilde v_1^2, (\tilde u \tilde v_1)^2 \rangle  \simeq \langle \tilde v_1 \; | \; \tilde v_1^2\rangle \times \langle \tilde u \tilde v_1 \; | \; (\tilde u \tilde v_1)^2\rangle.
\end{equation}
For ${\rm Dih}_\infty \times {\rm Dih}_1$ isomorphism class, we have 
\begin{equation}
\begin{aligned}
\tilde F & = \langle \tilde u,\tilde v_1,\tilde v_2 \; | \; \tilde u^2,  \tilde v_j^2,  (\tilde v_1 \tilde v_2)^2, (\tilde u \tilde v_j)^2\rangle  \\
& \simeq \langle \tilde v_1 \; | \; \tilde v_1^2\rangle \times \langle \tilde v_2 \; | \; \tilde v_2^2\rangle \times \langle \tilde u \tilde v_1 \; | \; (\tilde u \tilde v_1)^2\rangle.
\end{aligned}
\end{equation}
and the statement follows again.
\end{proof}

\fi

For us, the relevant finite group is $\tilde F : = F/2{\mathbb Z}$. Now, given a finite group $G$, any of its finite-dimensional representations supply a morphism $C^\ast_r(G) \to {\mathbb K}$, hence a cycle from $KK\big (C^\ast_r(G),{\mathbb C}\big )$. On the other hand, the $K_0$-theory of $C^\ast_r(G)$ can be read off from the ring of representations of the group $G$ \cite{LueckStamm}[Th.~3.2]. Then we have canonical pairings $KK\big ({\mathbb C},C^\ast_r(G)\big ) \times KK\big (C^\ast_r(G),{\mathbb C}\big ) \to {\mathbb Z}$, which in the present context work as follows:

\begin{proposition}
\label{prop:K0labeling}
Let $F$ be any of the frieze groups and $\tilde F$ be as above. If $\tilde p$ is a projection from $M_N({\mathbb C}) \otimes C^\ast_r(\tilde F)$, then the map
\begin{equation}\label{Eq:Rep0}
\tilde F \ni \tilde f \mapsto \left .U_{\tilde f}\right |_{\pi(\tilde p) [{\mathbb C}^N \otimes \ell^2(\tilde F)]}
\end{equation}
supplies a representation of $\tilde F$ by linear maps on the invariant sub-space $\pi(\tilde p) [{\mathbb C}^N \otimes \ell^2(\tilde F)]$. Let $\{\rho_a\}_{a = \overline {1,|\tilde F|}}$ be a complete set of irreducible representations of $\tilde F$. Then $\{\rho_a\}$ extend to morphisms from ${\mathbb K} \otimes C^\ast_r(\tilde F)$ to ${\mathbb K}$ that generate a basis for Kasparov's group $KK\big (C^\ast_r(\tilde F), {\mathbb C}\big )$. Furthermore, the pairings
\begin{equation}\label{Eq:Pairing1}
\tilde n_a : = [\tilde p]_0 \times [\rho_a] = \frac{1}{|\tilde F|}\sum_{\tilde f \in \tilde F}  {\rm Tr}\big (U_{\tilde f} \pi(\tilde p)\big ) \, {\rm Tr}\big (\rho_a(\tilde f)\big ) \in {\mathbb Z},
\end{equation}
supply a full set of $K$-theoretic labels for $\tilde p$.
Above, ``$\times$'' refers to the Kasparov product
\begin{equation}
KK\big ({\mathbb C},C^\ast_r(\tilde F)\big ) \times KK\big (C^\ast_r(\tilde F), {\mathbb C}\big ) \to KK({\mathbb C},{\mathbb C}) \simeq {\mathbb Z}.
\end{equation}
\end{proposition} 

\begin{proof} Any projection $\tilde p$ can be seen as a $C^\ast$-algebra morphism from ${\mathbb C}$ to $M_N({\mathbb C}) \otimes C^\ast_r(\tilde F)$. Indeed, the map
\begin{equation}
{\mathbb C} \ni c \mapsto c \, \tilde p \in M_N({\mathbb C}) \otimes C^\ast_r(\tilde F)
\end{equation}
preserve multiplication precisely because of the idempotency of $\tilde p$,
\begin{equation}
c_1 c_2 \mapsto (c_1 \tilde p) (c_2 \tilde p) =c_1 c_2 \, \tilde p^2 = c_1 c_2 \, \tilde p.
\end{equation}
We denote this morphism by the same symbol $\tilde p$. We claim that Eq.~\eqref{Eq:Pairing1} is nothing else but the trace of $({\rm id} \otimes \rho_a) \circ \tilde p$. According to our observation at the beginning of the previous subsection, $({\rm id} \otimes \rho_a) \circ \tilde p$ encodes the Kasparov product $[\tilde p]_0 \times [\rho_a]$, which lands in $KK_0({\mathbb C},{\mathbb C})$, and the trace applied on $({\rm id} \otimes \rho_a) \circ \tilde p$ is simply the isomorphism mapping $KK_0({\mathbb C},{\mathbb C})$ to ${\mathbb Z}$. Thus, the proof is complete if we can confirm the above claim. Now, suppose first that $\tilde p$ is from $C^\ast_r(\tilde F)$. Then, if $\tilde p = \sum_{\tilde f} \tilde p_{\tilde f} \, \tilde f$, the coefficients of this expansion are given by (see Remark~\ref{Re:TT})
\begin{equation}
\tilde p_{\tilde f} = {\mathcal T} (\tilde f^{-1} \, \tilde p) = \frac{1}{|\tilde F|}{\rm Tr} \big (U_{\tilde f} \, \pi(\tilde p) \big ).
\end{equation}
More generally, if $\tilde p$ is from $M_N({\mathbb C}) \otimes C^\ast_r(\tilde F)$, then the coefficients $\tilde p_{\tilde f}$ are from $M_N({\mathbb C})$ and $\tilde p_{\tilde f} = ({\rm id} \otimes {\mathcal T}) (\tilde f^{-1}\tilde p)$. Consequently, 
\begin{equation}
{\rm Tr}(\tilde p_{\tilde f}) = \frac{1}{|\tilde F|}{\rm Tr} \big (U_{\tilde f} \, \pi(\tilde p) \big ).
\end{equation}
Then
\begin{equation}
{\rm Tr}\Big (({\rm id} \otimes \rho_a)(\tilde p) \Big ) = {\rm Tr}\Big (\sum_{\tilde f \in \tilde F} \tilde p_{\tilde f} \otimes \rho_a(\tilde f) \Big ) = \frac{1}{|\tilde F|}\sum_{\tilde f \in \tilde F} {\rm Tr}(U_{\tilde f} \pi(\tilde p)\big ) {\rm Tr}\big (\rho_a(\tilde f) \big ),
\end{equation}
and this completes the proof.
\end{proof}

\begin{remark}{\rm The representations $\rho_a$ generate a complete set of mutually orthogonal central projections \cite{SerreBook}[pg.~50]
\begin{equation}
	\tilde p_a = \tfrac{d_a}{|\tilde F|} \sum_{\tilde f \in \tilde F} {\rm Tr}\big (\rho_a(\tilde f)\big ) \, \tilde f^{-1}, \quad d_a ={\rm dim} \, \rho_a,
\end{equation} 
for which
\begin{equation}
	C^\ast_r(\tilde F) = \bigoplus_a \tilde{p}_a C^\ast_r(\tilde F) \tilde{p}_a \simeq \bigoplus_a \rho_a(C^\ast_r(\tilde F)) =  \bigoplus_a M_{N_a}({\mathbb C})
\end{equation}
and a basis of $K_0(C^\ast_r(\tilde F))$ is obtained by choosing a rank-$1$-projection from each block. The K-theoretic labels $\tilde n_a$ in Eq.~\eqref{Eq:Pairing1} are the expansion coefficients of a projection with respect to such a basis.} $\Diamond$
\end{remark}  


\begin{remark}{\rm The right side of Eq.~\eqref{Eq:Pairing1} is the standard relation in representation theory, returning the power $n_a$ in the decomposition $\oplus_a \; \rho_a^{n_a}$ of the representation~\eqref{Eq:Rep0} in the irreducible representations. The Kasparov groups and product are used here to place these algebraic relations in a topological context. Indeed, when phrased as above, the pairings communicate much more, specifically, that the $K$-theoretic labels are invariant against continuous stable deformations of $\tilde p$. The Kasparov groups and product also supplies a very convenient book keeping instrument. Lastly, they enable us to place this and the following computations in a framework that can supply guidance in other situations, such as when the space groups in two and three dimensions are considered.
}$\Diamond$
\end{remark}

Now, according to Proposition~\ref{Pro:PhiKPiK}, $\varphi_2$ lands exactly in the right regular representation of $\tilde F$. Hence $\varphi_2$ can be seen as a cycle from $KK\big (C^\ast_r(F), C^\ast_r(\tilde F)\big )$. The following statement, together with the explicit checks of the assumptions, carried out in the next subsection, assures us that $\varphi_2$ is the actual sought cycle, mentioned in our discussion of the general strategy:

\begin{theorem}\label{Th:Main} Let $F$ be any of the frieze groups and $\{p_a\}$ be the basis of the $K_0$-group of $C^\ast_r(F)$, as listed in the previous section. If there is a complete set of irreducible representations $\{\rho_b\}$ of $\tilde{F}=F/2{\mathbb Z}$ such that the map,
\begin{equation}\label{Eq:PreTI}
K_0\big (C^\ast_r(F) \big ) \ni [p_a]_0 \mapsto \big \{[p_a]_0 \times [\varphi_2] \times [\rho_b] \big \} \in {\mathbb Z}^{\times |\tilde F|},
\end{equation}
is injective, then we can find a set of representations $\{\chi_a\}$ and corresponding classes $\{[\chi_a]\} \subset KK\big (C^\ast_r(\tilde F), {\mathbb C}\big )$ resolving the K-theoretic labels, in the sense that
\begin{equation}
[p_a]_0 \times [\varphi_2] \times [\chi_b] = \Lambda \delta_{a,b}, \quad \Lambda \in {\mathbb N}^\times.
\end{equation}
Consequently, the full set of K-theoretic labels of a projection $p$ from ${\mathbb K} \otimes C^\ast_r(F)$ can be computed as
\begin{equation}\label{Eq:TI}
n_a = \frac{1}{\Lambda |\tilde F|}\sum_{\tilde f \in \tilde F}  {\rm Tr}\big(U_{\tilde f} \varphi_2(p)\big) \, {\rm Tr}\big(\chi_a(\tilde f)\big ) \in {\mathbb Z}.
\end{equation}
\end{theorem}

\begin{proof}
The labels of Proposition~\ref{prop:K0labeling} can be grouped into a matrix ${\rm R}$ with integer entries ${\rm R}_{ba}= \big([p_a]_0 \times [\varphi_2]\big) \times [\rho_b]$, describing a homomorphism $K_0(C_r^*(F))\simeq {\mathbb Z}^{{\mathcal N}} \to {\mathbb Z}^{\lvert \tilde{F}\rvert}$, where ${\mathcal N}$ is the dimension of the corresponding $K_0$-group. By assumption on the injectivity, ${\rm R}$ has a left inverse $M$ with rational entries and, as such, we can find $\Lambda \in {\mathbb N}^\times$ and a matrix $\tilde M$ with integer entries such that $\sum_c \tilde M_{bc} {\rm R}_{ca} = \Lambda \, \delta_{ab}$. Setting
\begin{equation}\label{Eq:ChiTilde}
\tilde \chi_b(f) = \Lambda^{-1} \sum_{c} {\rm Tr}(\rho_c(f)) \tilde M_{bc} = \sum_{c} {\rm Tr}(\rho_c(f)) M_{bc},
\end{equation}
then, for any projection $p \sim p_a^{\otimes n_a}$, one has
\begin{align*}
\frac{1}{|\tilde F|}\sum_{\tilde f \in \tilde F}  {\rm Tr}\big(U_{\tilde f} \varphi_2(p)\big) \, \tilde \chi_b(\tilde f) &= \frac{1}{|\tilde F|} \sum_{\tilde f \in \tilde F} n_a  {\rm Tr}\big(U_{\tilde f} \varphi_2(p_a)\big) \, \tilde \chi_b(\tilde f)\\
&= \frac{1}{|\tilde F|} \sum_{\tilde f \in \tilde F} n_a  {\rm Tr}\big(U_{\tilde f} \varphi_2(p_a)\big) \, \tilde \chi_b(\tilde f)\\
&= \frac{1}{|\tilde F|} \sum_{\tilde f \in \tilde F} n_a  {\rm Tr}\big(U_{\tilde f} \varphi_2(p_a)\big) \, \sum_{c} {\rm Tr}(\rho_c(f)) M_{bc}
\\
&=\sum_{c} n_a  {\rm R}_{ca}  M_{bc} = n_a \delta_{ab}.
\end{align*}
Lastly, if we set $\chi_b = \oplus_c \rho_c^{\tilde M_{bc}}$ as the desired representations, then 
\begin{equation}
\tilde \chi_b(\tilde f) = \Lambda^{-1} \, {\rm Tr}\big(\chi_b(\tilde f)\big )
\end{equation}
and the statement follows. \end{proof}

\section{Numerical Experiments}
\label{Sec:Numerical}

\subsection{The generating dynamical matrices} One refers to a set of dynamical matrices as generating models if their band projections supply a complete basis of the $K_0$-group. Of course, one can can always start from $h_j = -p_j$ with $\{p_j\}$ a basis of the $K_0$-group, but sometime this fails because $p_j$'s may involve an infinite number of coefficients in their expansion, {\it i.e.} they don't have  finite range. The latter is essential for laboratory implementations of the models.  While this will not be the case here, let us recall that, in such cases, one simply truncates the series, making sure that the truncation does not close the spectral gap. This opening paragraph is a brief reminder of the well known fact that a successful computation of the $K_0$-group and of its generators is equivalent to finding all relevant topological models supported by the algebra of dynamical matrices. Any other dynamical matrix can be deformed into a  stack of elementary models from this generating set.

\vspace{0.1cm}

The present context is special because all model Hamiltonians can be generated directly from the basis of the $K_0$-groups. For example, for the group $p2mm$, the generating dynamical matrices are $h_j = -p_j$, where $p_j$'s are the projections listed in Eq.~\eqref{Eq:K0Generators}. Explicitly,
\begin{equation}\label{Eq:GenModels1}
  h_1 = -e, \ h_2 = \tfrac{1}{2}(v_1-e), \ h_3 = \tfrac{1}{2}(u v_1-e), \ h_4 = -\tfrac{1}{2}(e+v_2)
  \end{equation}
 and
  \begin{equation}\label{Eq:GenModels2}
 h_5 = \tfrac{1}{4}(v_1 v_2-v_1 -v_2 +e), \ h_6= \tfrac{1}{4}(u v_1 v_2-uv_1 -v_2 +e)
 \end{equation}
 
 \begin{figure}
\center
  \includegraphics[width=0.99\linewidth]{TermActions.pdf}
  \caption{\small The right regular representations of key elements from $C^\ast_r(F)$, $F = p2mm$. The resonators of the patterns are labeled by the group elements of $p2mm$ and a line between two resonators labeled by $f$ and $f'$ represents the hopping operator $|f\rangle \langle f'|$. The right regular representations are the sums of hopping operators displayed in the corresponding panel. In the top-left panel, the elements and the corresponding connections are color coded.}
 \label{Fig:TermActions}
\end{figure}

\vspace{0.1cm}

Their right regular representations can be computed by repeatedly applying the following rule
\begin{equation}
\pi(u^k v^\beta) | u^n v^\alpha \rangle = |u^{n-(-1)^{\alpha_1+\beta_1}k}v^{\alpha+\beta} \rangle.
\end{equation}
These same rules can be easily adopted for the other isomorphism classes of frieze groups, by simply restricting the range of the $\alpha$ coefficients. In fact, focusing on the group $p2mm$ is enough because all the other groups can be embedded in it. In particular, we can list the generating models for the other isomorphism classes: $\{h_1\}$ for $Z_\infty$; $\{h_1,h_2,h_3\}$ for ${\rm Dih}_\infty$; $\{h_1,h_4\}$ for $Z_\infty \times {\rm Dih}_1$.

\vspace{0.1cm}

For the reader's convenience, we report in Fig.~\ref{Fig:TermActions} a graphical illustration of the right regular representations of some key algebra elements that cover all first nearest-neighbor couplings. These diagrams can guide the experimental scientists on how to create model dynamical matrices that belong to a specific topological class. For example, after examining the expressions of the projections~\eqref{Eq:K0Generators}, one should conclude that making the blue couplings in the top-left panel dominant will produce a dynamical matrix from the class of $h_3$. Similarly, for $h_2$, and $h_4$. For a dynamical matrix from the topological class of $h_5$, one should include the couplings shown in the bottom panels and in the top-right panel, hence three types of connections of approximately equal strength. For guidance, we exemplify these ideas in Fig.~\ref{Fig:ModelHamP}. Realistic resonating structures will be supplied in section~\ref{Sec:PlateResonators}. The Mathematica code used to generate Fig.~\ref{Fig:ModelHamP} is available from https://www.researchgate.net/profile/Emil-Prodan.

\subsection{Numerical algorithm for the K-theoretic labels}\label{Sec:NA} As we argued in the first section, a dynamical matrix is simply a self-adjoint element $h$ from $M_N({\mathbb C}) \otimes C^\ast_r(F)$ and so are its spectral projections. However, these elements connect with the physical models through the left regular representations, hence we want our numerical experiments to be carried out within these representations. The group $F$, however, will be reduced to $F/k{\mathbb Z}$, for some large even integer $k$, in order to generate a finite system with no boundary. As we already stated, the subgroup $k{\mathbb Z}$ is normal in $F$, hence the quotient of a self adjoint element $h$ from $F$ to $F/k{\mathbb Z}$ preserves the resolvent spectrum
\begin{equation}
{\mathbb R} \setminus {\rm Spec}(h) \subseteq {\mathbb R} \setminus {\rm Spec}(h \cdot k{\mathbb Z}).
\end{equation}
In other words, the spectral gaps of $h$ are not contaminated by our finite volume approximations, as it is usually the case if other boundary conditions are imposed.

 \begin{figure}
\center
  \includegraphics[width=0.99\linewidth]{PatternHamiltonians.pdf}
  \caption{\small Generating dynamical matrices produced from actual patterns, with the resonators assumed to interact as $e^{-2|f\cdot 0-f'\cdot 0|}|f\rangle \langle f'|$, {\it i.e.} with a coupling strength that decays exponentially with the distance between their centers. The values of the six topological invariants associated with the bottom spectral band are displayed in each panel.}
 \label{Fig:ModelHamP}
\end{figure}

\vspace{0.1cm}

First, we need to acknowledge a few practical book keeping details. The group $F/k {\mathbb Z}$ is manipulated on the computer by mapping it in a convenient model group. This model group is simply a subgroup of the permutation group of $|F/k {\mathbb Z}|$ objects. For $F=p2mm$, for example, the group homomorphism is implemented by labeling the group elements as
\begin{equation}
 u^n v_1^{\alpha_1} v_2^{\alpha_2} \mapsto f_j, \quad j = 4n+2\alpha_2 + \alpha_1+1, \quad n = \overline{0,k-1}, \quad \alpha_j =0,1,
\end{equation}
which gives rise to a map $F/k {\mathbb Z} \to \{1,2,\ldots, |F/k {\mathbb Z}|\}$. Then the group table can be encoded in a two-index array 
\begin{equation}\label{Eq:Prod}
(j,j') \mapsto {\rm Prod}_{j,j'} \in \{1,\ldots, |F/k {\mathbb Z}|\},
\end{equation} 
by using the rule~\eqref{Eq:Compo1}. Obviously, the relation stated in Eq.~\eqref{Eq:Prod} assigns a unique permutation to each element $f_j \in F/k{\mathbb Z}$, acting as
\begin{equation}
f_j(j') ={\rm Prod}_{j,j'}, \quad j' \in \{1,\ldots,|F/k{\mathbb Z}|\}.
\end{equation}

The basis $|i,f\rangle$ of the Hilbert space corresponding to the right regular representation of $F/k {\mathbb Z}$ is labeled using the scheme
\begin{equation}
|i,f_j\rangle \mapsto | w \rangle, \quad w=i+ N_s j, \quad i=\overline{1,N_s}, \quad j=\overline{1,|F/k{\mathbb Z}|},
\end{equation}
and the matrix elements of the representations $\pi(f)$ are stored using the rule~\eqref{Eq:MatElem}, which is applied using the labels and the array ${\rm Prod}$ introduced above. In both rules \eqref{Eq:Compo1} and \eqref{Eq:MatElem}, the sums involving the index $n$ are computed modulo $k$. By setting $k=2$, the above schemes can and are applied to the group $\tilde F$ too.

\vspace{0.1cm}

Now, the first challenge for us is how to implement the map $\varphi_2$. The solution rests on using the spectral projections of the reduced element $u \in F/k{\mathbb Z}$:
\begin{equation}
P_j = \tfrac{1}{k} \sum_{n =0}^{k-1} e^{-\imath 2 \pi j n/k} \pi(u)^n, \quad j=\overline{0,k-1}.
\end{equation}
Then we define $\varphi_2$ by projecting
\begin{equation}\label{Eq:Phi2}
F/k {\mathbb Z} \ni f \mapsto \varphi_2(f) = \Pi \pi(f) \Pi, \quad \PI : = P_0 \oplus P_{k/2}.
\end{equation} 
Eq.~\eqref{Eq:Phi2} is justified by the following relations:
\begin{equation}
\pi(u^m) \Pi = \Pi \pi(u^m)= \pi(u)^{m \, {\rm mod} 2} \, \Pi, \quad \pi(v_j) \Pi = \Pi \pi(v_j), 
\end{equation}
which enable us to evaluate
\begin{equation}
\Pi \sum_\alpha \sum_{n=0}^{k-1} c(n,\alpha) \pi\big (u^n v^\alpha \big)  = \Pi \sum_{m=0,1}\sum_{\alpha}\Big (\sum_{r \in 2{\mathbb Z}} c(m+r,\alpha)\Big ) \pi(u)^m \pi(v)^\alpha 
\end{equation}
and confirm that Eq.~\ref{Eq:Phi2} is equivalent with Eq.~\ref{Eq:PhiE}, once we make the identifications 
\begin{equation}
\tilde u : = \Pi \pi(u) \Pi, \quad \tilde v_j : = \Pi \pi(v_j) \Pi.
\end{equation}

\vspace{0.1cm}

\begin{figure}
\center
  \includegraphics[width=0.99\linewidth]{GeneratingModels.pdf}
  \caption{\small Spectrum of perturbations of the generating dynamical matrices listed in Eqs.~\eqref{Eq:GenModels1} and \eqref{Eq:GenModels2}, together with the topological invariants $n_a$, $a=\overline{1,6}$, as computed with Eq.~\eqref{Eq:NTI1} for the gap projections.}
 \label{Fig:GenModels}
\end{figure}

The second and last challenge is how to evaluate the trace ${\rm Tr}\big(U_{\tilde f} \varphi_2(p)\big)$. Switching the order of the operators inside the trace, we have
\begin{equation}
{\rm Tr}\big(\varphi_2(p) U_{\tilde f} \big) = \sum_{i=1}^{N_s}\sum_{\tilde f' \in \tilde F} \langle i, \tilde f' |  \varphi_2(p) U_{\tilde f} |i, \tilde f'\rangle = \sum_{i=1}^{N_s} \sum_{\tilde f' \in \tilde F}\langle i, \tilde f' | \varphi_2(p) |i,\tilde f \tilde f' \rangle.
\end{equation}
Next, we connect with what was said above and represent the elements $\tilde f'$ of $\tilde F$ as $\Pi f$ with $f$ from $F$, in which case,
\begin{equation}
{\rm Tr}\big(U_{\tilde f} \varphi_2(p)\big) = \sum_{i=1}^{N_s} \sum_{f \in F/k{\mathbb Z}}\langle i, f | \Pi \pi(p) \Pi |i,\tilde f f \rangle,
\end{equation}
where the sum is taken over all elements of $F/k{\mathbb Z}$. Here, $\tilde f \in F/2 {\mathbb Z}$ is seen as an element of $F/k {\mathbb Z}$ via the canonical section of the morphism $F/k {\mathbb Z} \to F/2 {\mathbb Z}$. To verify that the normalization constants are properly fixed, we have verified that ${\rm Tr}\big(U_{\tilde 1} \varphi_2(1)\big)=8$, the dimension of the right regular representation of $\tilde F$ when $N_s$ is set to one. 

\vspace{0.1cm}

We now have the numerical tools to examine the map defined in Eq.~\eqref{Eq:PreTI}. We use the numerical formula
\begin{equation}\label{Eq:NTI1}
[p_a]_0 \mapsto \frac{1}{|\tilde F|}\sum_{\tilde f \in \tilde F} \rho_b(\tilde f) \sum_{i=1}^{N_s} \sum_{f \in F/k{\mathbb Z}}\langle i, f | \Pi \pi(p_a) \Pi |i, \tilde f f \rangle
\end{equation}
and the eight independent one dimensional irreducible representations of $\tilde F$ enumerated below: 
\begin{equation}
\rho : =\Big [\rho_a (\tilde f_b) \Big ]_{a,b=\overline{1,8}} = {\tiny \begin{pmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\

 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 \\

 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 \\

 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 \\

 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 \\

 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 \\

 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\

 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1  \end{pmatrix} }.
\end{equation} 
They come from the presentation of $\tilde F$ as the direct product $\langle \tilde v_1 \rangle \times \langle \tilde v_2 \rangle \times \langle \tilde u \rangle$ and from the tensor products of the standard representations of the constituent ${\mathbb Z}_2$ subgroups. In particular, all $\rho_a$ representations are one dimensional. As a check, we can confirm that $\rho \cdot  \rho^T = I_{8 \times 8}$. 

\vspace{0.1cm}

The result for ${\rm R}$ is
\begin{equation}
\big [{\rm R}_{ba}\big] = \Big [[p_a]_0 \times [\varphi_2] \times [\rho_b] \Big ]_{^{b=\overline{1,8}}_{a=\overline{1,6}}}= {\tiny \begin{pmatrix} 
1 & 0 & 0 & 1 & 0 & 0 \\
 1 & 1 & 1 & 1 & 0 & 0 \\
 1 & 0 & 1 & 1 & 0 & 0 \\
 1 & 1 & 0 & 1 & 0 & 0 \\
 1 & 0 & 0 & 0 & 0 & 0 \\
 1 & 1 & 1 & 0 & 1 & 1 \\
 1 & 0 & 1 & 0 & 0 & 1 \\
 1 & 1 & 0 & 0 & 1 & 0
 \end{pmatrix}}
\end{equation}
and the rank of this matrix is six, hence equal to the dimension of the $K_0$-group. This confirms that the assumption in Theorem~\ref{Th:Main} holds. It also reassures us that the six generators $\{p_a\}$ listed in Eq.~\eqref{Eq:K0Generators} are indeed independent. Furthermore, we can produce the left inverse $M=\big ( {\rm R}^T \cdot {\rm R} \big )^{-1} \cdot {\rm R}^T$ of ${\rm R}$ and define the linear maps $\{\tilde \chi_a\}$ introduced in Eq.~\eqref{Eq:ChiTilde},
\begin{equation}
\Big [\tilde \chi_a(\tilde f_b)\Big ]_{^{a=\overline{1,6}}_{b=\overline{1,8}}} = M \cdot \rho = {\tiny \begin{pmatrix}
1 & 1 & -1 & -1 & 0 & 1 & 0 & -1 \\

 0 & -2 & 0 & -2 & 0 & 0 & 0 & 0 \\

 0 & 0 & 0 & 0 & 0 & -2 & 0 & -2 \\

 0 & 0 & 2 & 2 & 0 & 0 & 0 & 2 \\

 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 \\

 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4
 \end{pmatrix} },
 \end{equation}
which deliver the desired pairings. Lastly, the K-theoretic labels of a projection $p$ are computed numerically via
\begin{equation}\label{Eq:NTI1}
n_a = \frac{1}{|\tilde F|}\sum_{\tilde f \in \tilde F} \tilde \chi_b(\tilde f) \sum_{i=1}^N \sum_{f \in F/k{\mathbb Z}}\langle i, f | \Pi \pi(p) \Pi |i, \tilde f f \rangle .
\end{equation}

\begin{figure}
\center
  \includegraphics[width=0.99\linewidth]{Interpolation.pdf}
  \caption{\small Demonstrations of topological spectral flow generated by linear interpolation between models displaying  bands with distinct K-theoretic labels. The spectra of the interpolated dynamical matrices are shown on the right, together with the six topological invariants of the seen spectral bands.}
 \label{Fig:Inter}
\end{figure}

\vspace{0.1cm}

We want to point out that this numerical formula can be canonically adapted to space groups in arbitrary dimensions, once the finite subgroup is identified.

\subsection{Computations of K-theoretic labels} We first validate our numerical algorithm by computing the K-theoretic labels for the band projections of the model dynamical matrices listed in Eqs.~\eqref{Eq:GenModels1} and \eqref{Eq:GenModels2}. To make it somewhat more challenging, we actually added a perturbation $\delta h$, which was generated by populating the series~\eqref{Eq:Elem1} with random entries up to  $|n| \leq 2$ (hence 24 random terms). Fig.~\ref{Fig:GenModels} reports the spectra of these dynamical matrices, together with evaluations of the parings~\eqref{Eq:NTI1} for the gap projections corresponding to the indicated spectral gaps. As one can see, the numerical results confirm the statement made in Eq.~\eqref{Eq:TI}. The Mathematica code used to generate Fig.~\ref{Fig:GenModels} is available from https://www.researchgate.net/profile/Emil-Prodan.

\vspace{0.1cm}

Next, we want to demonstrate observable physical effects predicted with the tools developed by our work. For this, let $e_{i}$ be the matrix with entry 1 at position $(i,i)$. We then generate Hamiltonians from $M_6({\mathbb C}) \otimes C^\ast_r(F)$ using the scheme
\begin{equation}\label{Eq:MultiLayer}
h = \sum_{i=1}^6 \Lambda_i \; e_i \otimes h_{\xi(i)}
\end{equation}
where the coefficients $\Lambda_i$ are chosen such that $h$ displays a desired number of bands and $\xi$ is a permutation of the indices, which will enable us to shuffle the bands and to engineer interesting K-theoretic labels.

\vspace{0.1cm}

Fig.~\ref{Fig:Inter} reports spectra of dynamical matrices generated with the scheme outlined above, together with the computed K-theoretic labels. Fig~\ref{Fig:Inter} also illustrates how a linear interpolation between models with different K-theoretic labels prompts the predicted topological spectral flow. The Mathematica code used to generate Fig.~\ref{Fig:Inter} is available from https://www.researchgate.net/profile/Emil-Prodan.

\section{Topological spectral flow with plate resonators}
\label{Sec:PlateResonators}

\begin{figure}
\center
  \includegraphics[width=0.99\linewidth]{BasicExpDesign.pdf}
  \caption{\small (a) A resonating structure of coupled steel plates, generated with the frieze group $p2mm$. (b) The seed resonator, together with its physical characteristics and a representation of its first 10 resonant modes. (c) Details of the couplings, notations and values of the fixed parameters.}
 \label{Fig:BD1}
\end{figure}

In this section, we illustrate how the discrete models discussed so far can be implemented with physical resonating structures. Fig.~\ref{Fig:BD1}(a) describes a pattern of coupled steel plates, generated with the frieze group $p2mm$ from one seed resonator. The basic plate resonator and its physical characteristics are presented in Fig.~\ref{Fig:BD1}(b), together with its first ten resonant modes. We focus on the dynamical features generated from the first flexural mode of the plate, indicated in Fig.~\ref{Fig:BD1}(b) by the star. Note that the geometry of the plate was chosen such that this mode is well separated in frequency from all the other resonant modes. As such, the collective dynamics of the plates in the regime of weak couplings and in a frequency domain around the frequency of this particular mode can be accurately quantified using a discrete model with one resonant mode per resonator.

\vspace{0.1cm}

 Couplings between the resonators are implemented by thin metal bridges that respect the $p2mm$ symmetry. The strength of these couplings can be controlled by the position of these bridges: A bridge placed closer to a node (anti-node) of the resonant mode produces a weaker (stronger) coupling (see the mode's spatial profile in Fig.~\ref{Fig:BD1}(b)). The strength of the couplings can be further adjusted by modifying the Young modulus $E$ of the metal bridges. Thus, our space of adjustable parameters consists of five parameters $(h_1,h_2,h_3,E_1,E_2)$ (see Fig.~\ref{Fig:BD1}(c)).

\begin{figure}
\center
  \includegraphics[width=0.5\linewidth]{TenClasses.pdf}
  \caption{\small Configurations of the couplings generating dynamical matrices from the topological class of the algebra elements mentioned in Fig.~\ref{Fig:TermActions}. The Young modulus of the metal bridges was fixed at $E_1 = E_2=0.1$GPa, except for the two cases indicated in the diagram. The diagram also shows and labels the ten distinct ways in which the physical systems can be interpolated.}
 \label{TenClass}
\end{figure}

\vspace{0.1cm}

In Fig.~\ref{Fig:ModelHamP}, we indicated how the resonators are to be coupled in order to reproduce the fundamental model dynamical matrices. However, in practice, we found that it is much easier to implement dynamical matrices $K_i$, $i = \overline{1,5}$, from the topological classes of the self-adjoint elements presented in Fig.~\ref{Fig:TermActions}. Indeed, for each of those elements, we only need to ensure one specific dominant coupling and this can be achieved with the configurations shown in Fig.~\ref{TenClass}. We have computed the K-theoretic labels of the lowest spectral bands of $K_i$ dynamical matrices, based on the assumption that they indeed belong to topological class of the algebra elements indicated in Fig.~\ref{TenClass}. The results are: $K_1 \rightarrow (0,1,0,0,0,0)$, $K_2 \rightarrow (0,0,1,0,0,0)$, $K_3 \rightarrow (1,0,0,-1,0,0)$, $K_4 \rightarrow (1,1,0,-1,-2,0)$, $K_5 \rightarrow (1,0,1,-1,0,-2)$,
 hence they are all distinct from each other. The Mathematica code used to generate these topological labels is available from https://www.researchgate.net/profile/Emil-Prodan.
 
\vspace{0.1cm}

\begin{remark}{\rm Among other things, the above K-theoretic labels assures us that actually $K_i$ together with $K_0 = 1$ can serve as an alternative generating set of model dynamical matrices. In many respects, this set is more natural than the one we originally considered.
}$\Diamond$
\end{remark}

Each of the resonating structures seen in Fig.~\ref{TenClass} corresponds to a point in the 5-dimensional parameter space $(h_1,h_2,h_3,E_1,E_2)$. Based on the K-theoretic labels computed above, we predict that, no matter how we interpolate between two of these five points, we cannot avoid the closing of the gap in the resonant spectrum. This phenomenon should be extremely robust because the K-theory predicts that the gap closing cannot be avoided even if the structure is connected to external frames, provided this connection does not change the topological class of the initial and final resonating structures. Furthermore, we conjecture that the topological spectral flows are robust against disorder.

\vspace{0.1cm}

The five topologically distinct resonating structures from Fig.~\ref{TenClass} can be pairwise interpolated in ten different ways, as specified there. Each of these interpolations will produce distinct topological spectral flows that can find different practical applications. In Fig.~\ref{Fig:ESF}, we report the COMSOL-simulated resonant spectra for the ten interpolations between different $(i,j)$ pairs from Fig.~\ref{TenClass}, where the parameters were varied linearly as
\begin{equation}\label{Eq:Interp}
\begin{aligned}
\big 
(h_1(s) ,h_2(s), h_3(s), E_1(s), E_2(s)\big)  = & (1-s)(h_1^i ,h_2^i, h_3^i, E_1^i, E_2^i) \\
& +s(h_1^j ,h_2^j, h_3^j, E_1^j, E_2^j).
\end{aligned}
\end{equation}
These simulations fully confirm the K-theoretic predictions and serve as a demonstration of how spectral engineering can be achieved with plate resonators generated by space groups.

\begin{figure}
\center
	\includegraphics[width=0.99\linewidth]{ExpSpecFlow.pdf}
	\caption{\small (a) Spectral flow of the resonant spectrum when the configuration of the coupled plate resonators is deformed (see Eq.~\eqref{Eq:Interp}) along the ten distinct interpolations shown in Fig.~\ref{TenClass}.}
	\label{Fig:ESF}
\end{figure}






