# Discrete constant mean curvature cylinders and isothermic tori

Joseph Cho  Institute of Discrete Mathematics and Geometry, TU Wien, Wiedner
Hauptstrasse 8-10/104, 1040 Wien, Austria<EMAIL_ADDRESS>,
Katrin Leschke  Department of Mathematics, University of Leicester, University
Road, Leicester LE1 7RH, United Kingdom<EMAIL_ADDRESS>and  Yuta
Ogata  Department of Mathematics, Faculty of Science, Kyoto Sangyo University,
Motoyama, Kamigamo, Kita-ku, Kyoto-City, 603-8555, Japan<EMAIL_ADDRESS>su.ac.jp

###### Abstract.

We consider the monodromy problem of Darboux transforms of discrete isothermic
surfaces using the integrable theory of discrete polarised curves. Then we
provide, for the first time, closed-form discrete parametrisations of discrete
isothermic cylinders, discrete constant mean curvature cylinders, and discrete
isothermic tori.

###### Key words and phrases:

discrete constant mean curvature surface, discrete isothermic surface,
discrete isothermic cylinder, discrete isothermic torus

###### 2020 Mathematics Subject Classification:

Primary 53A70; Secondary 53A10, 53A31.

##  1\. Introduction

Discrete differential geometry aims to examine discrete surfaces through
integrable systems technique (see, for example, [bobenko_discrete_2008]), and
has gained interest recently from both mathematical and applicational
viewpoints. However, a large number of results concerning structure-preserving
discretisation of surfaces concern their local behaviour. Here, we aim to
consider a global aspect of discrete surfaces, focusing on the case of
_discrete isothermic surfaces_.

Isothermic surfaces [bour_theorie_1862] constitute an integrable class of
surfaces [cieslinski_isothermic_1995] (see also [burstall_curved_1997]), and
admit spectral deformations [bianchi_ricerche_1905, bianchi_complementi_1906,
calapso_sulla_1903, calapso_sulle_1915] and _Darboux transformations_
[darboux_sur_1899-1]. In particular, Darboux transformations have been used to
construct various isothermic surfaces with topological constraints by
considering the _monodromy problem_ : notably, constant mean curvature (cmc)
cylinders called bubbletons can be obtained as Darboux transformations of
circular cylinders [bianchi_lezioni_1903] (see also
[sterling_existence_1993]), while non-trivial isothermic tori were found via
Darboux transforms of the homogeneous tori in $3$-sphere [bernstein_non-
special_2001].

On the other hand, _discrete isothermic surfaces_ were defined in
[bobenko_discrete_1996-1] via the permutability of Darboux transformations
[bianchi_ricerche_1905], and _Darboux transformations_ for discrete isothermic
surfaces were defined in [hertrich-jeromin_discrete_1999] using a discrete
Riccati-type equation that is equivalent to a cross-ratios conditon. A gauge
theoretic approach to discrete isothermicity and its Darboux transforms was
explored in both the quaternionic setup and the lightcone model [hertrich-
jeromin_transformations_2000, burstall_discrete_2014], where isothermicity is
characterised by a existence of a certain $1$-parameter family of discrete
flat connections on the discrete trivial bundle, with Darboux transforms given
by their parallel sections. One can also consider the integrable reduction to
the class of discrete cmc surfaces, so that certain Darboux transformations of
a given discrete cmc surface result in another discrete cmc surfaces
[hertrich-jeromin_discrete_1999, burstall_discrete_2014].

In this paper, we consider the monodromy problem for Darboux transformations
of discrete isothermic surfaces to construct discrete isothermic cylinders,
including discrete cmc cylinders, and discrete isothermic tori. In particular,
we are interested in obtaining _closed-form discrete parametrisations_ for
explicit examples of such surfaces; thus, we will use the quaternionic setup
for isothermicity, as this approach has proven to be a useful tool in
obtaining explicit solutions to Riccati-type equations in both the smooth and
discrete settings (see, for example, [cho_new_2022, cho_periodic_nodate]).

In Section 2, we first review the gauge theoretic description of discrete
isothermic surfaces and discrete polarised curves in the quaternionic setup;
we will also briefly review the theory of discrete isothermic cmc surfaces.
Then in Section 3, we view discrete isothermic surfaces as successive Darboux
transformations of discrete polarised curves (a discrete analogue of the
result obtained for semi-discrete isothermic surfaces in [burstall_semi-
discrete_2016]). This change in viewpoint is crucial for solving the monodromy
problem, as we show that the monodromy problem of Darboux transformations of
discrete isothermic surfaces can be reduced to that of discrete polarised
curves (see Theorem 3.5).

In Section 4, we consider Darboux transformations of explicit discrete
isothermic surfaces. After obtaining discrete isothermic cylinders as Darboux
transformations of the discrete circular cylinder in Section 4.3, we use the
fact that discrete circular cylinders are discrete cmc surfaces to find the
Darboux transforms that are discrete cmc cylinders in Section 4.4; these are
the discrete analogues of the bubbletons. Then we switch our attention to the
Darboux transformations of homogeneous tori in $3$-sphere, and obtain explicit
examples of discrete isothermic tori in Section 4.5. (See also Figure 1.)

Importantly, we provide, for the first time, _closed-form discrete
parametrisations_ of all the explicit examples we consider by solving the
discrete Riccati-type equation (see (4.6), (4.10), or (4.16)); the
explicitness of our formulation allows us to observe in Section 4.6 that under
the appropriate continuum limit, the closure of Darboux transforms is
preserved. Thus, our work suggests that structure preserving discretisation
can provide valuable insights in search of new examples of compact isothermic
surfaces.

Acknowledgements. The second author gratefully acknowledges the partial
support of this work provided by the Research Institute for Mathematical
Sciences, an International Joint Usage/Research Center located at Kyoto
University. The third author gratefully acknowledges the support from Grants-
in-Aid of JSPS Research Fellowships for Young Scientist 21K13799.

Figure 1. Discrete constant mean curvature cylinder (on the left) and discrete
isothermic torus (on the right) obtained as Darboux transformations of
discrete isothermic surfaces.

##  2\. Preliminaries

We review the quaternionic vector bundle formalism of Darboux transformations
of discrete isothermic surfaces and discrete polarised curves.

###  2.1. Discrete flat connections

First, we briefly review the vector bundle theory for discrete systems (see,
for example, [burstall_notes_2017, burstall_discrete_2023]). Let
$(m,n)\in\Sigma^{2}\subset\mathbb{Z}^{2}$ be a simply connected discrete
domain in the sense of [burstall_discrete_2023]. For the labeling of vertices
used throughout the paper, please refer to Figure 2. Writing $f(m,n)=f_{m,n}$
or $f(i)=f_{i}$ for any function $f$ defined on $\Sigma^{2}$, consider a
discrete bundle $W$, that is, an assignment of sets $V_{i}$ to each vertex
$i\in\Sigma^{2}$. We call
$\sigma\mathrel{\mathop{\ordinarycolon}}\Sigma^{2}\to\cup V_{i}$ a section,
i.e. $\sigma\in\Gamma W$, if we have $\sigma_{i}\in V_{i}$ for all $i$.

A discrete connection is an assignment of bijections
$r_{ji}\mathrel{\mathop{\ordinarycolon}}V_{i}\to V_{j}$ to each oriented edge
$(ij)$ so that $r_{ji}r_{ij}=\operatorname{id}_{j}$; discrete gauge
transformations $G$ act on the discrete connections $r_{ji}$ via

$(G\bullet r)_{ji}=G_{j}r_{ji}G_{i}^{-1}$

for automorphisms $G_{i}\mathrel{\mathop{\ordinarycolon}}V_{i}\to V_{i}$
defined on every vertex $i\in\Sigma$.

A connection is _flat_ if, on every elementary quadrilateral $(ijk\ell)$, we
have

$r_{i\ell}r_{\ell k}r_{kj}r_{ji}=\operatorname{id}_{i},$

and it follows that any gauge transform of a flat connection is flat. For a
flat connection, we call a section $r$-parallel if
$\sigma_{j}=r_{ji}\sigma_{i}$ on every edge $(ij)$.

$(m,n)$$(m,n+1)$$(m+1,n)$$(m,n+2)$$(m+1,n+1)$$(m+2,n)$$i$$\ell$$j$$p$$k$$q$
Figure 2. An elementary quadrilateral with auxiliary points and their labels.

###  2.2. Integrable structure of discrete isothermic surfaces

In this section. we review the basic integrable structure of discrete
isothermic surfaces in $\mathbb{H}\cong\mathbb{R}^{4}$, thoroughly
investigated in [hertrich-jeromin_transformations_2000]. While reviewing, we
will adapt the results of [hertrich-jeromin_transformations_2000] in terms of
discrete vector bundle theoretic language.

####  2.2.1. Discrete isothermic surfaces

For a simply connected $\Sigma^{2}\subset\mathbb{Z}^{2}$ with
$(m,n)\in\Sigma^{2}$, we first recall the definition of discrete isothermic
surfaces in the set of quaternions
$\mathbb{H}\mathrel{\mathop{\ordinarycolon}}=\operatorname{span}_{\mathbb{R}}\\{1,\mathbbm{i},\mathbbm{j},\mathbbm{k}\\}\cong\mathbb{R}^{4}$
given in [hertrich-jeromin_transformations_2000, Definition 3.2] (see also
[bobenko_discrete_1996-1, Definition 6]):

######  Definition 2.1.

A discrete surface $f\mathrel{\mathop{\ordinarycolon}}\Sigma^{2}\to\mathbb{H}$
is called _discrete isothermic_ if

$\operatorname{cr}(f_{i},f_{j},f_{k},f_{\ell})\mathrel{\mathop{\ordinarycolon}}=(f_{i}-f_{j})(f_{j}-f_{k})^{-1}(f_{k}-f_{\ell})(f_{\ell}-f_{i})^{-1}=\frac{\mu_{i\ell}}{\mu_{ij}}$
(2.1)

on every elementary quadrilateral $(ijk\ell)$ for some function $\mu$ defined
on unoriented edges satisfying the edge-labeling property, i.e.
$\mu_{ij}=\mu_{k\ell}$ and $\mu_{i\ell}=\mu_{jk}$. The function $\mu$ is
called the _cross-ratios factorising function_.

######  Remark 2.2.

The definition of isothermicity implies that the image of every elementary
quadrilateral $(ijk\ell)$ is _concircular_. Therefore, the class of discrete
isothermic surfaces is a special class of _circular nets_ , a discrete
analogue of curvature line parametrised surfaces [nutbourne_differential_1988,
p. 220]. For this reason, we will call $f_{m,n_{0}}$ or $f_{m_{0},n}$ for some
fixed $m_{0}$ or $n_{0}$ a _discrete $m$-curvature line_ or _discrete
$n$-curvature line_, respectively (see also, for example, [cho_discrete_2021,
Definition 2.7]).

For the discrete $1$-form $\omega_{ij}$ defined as

$\omega_{ij}=\frac{1}{\mu_{ij}}\operatorname{d\\!}{f}_{ij}^{-1}$

on every edge $(ij)$ where

$\operatorname{d\\!}{f}_{ij}\mathrel{\mathop{\ordinarycolon}}=f_{i}-f_{j},$

we have from [hertrich-jeromin_transformations_2000, Definition 3.4] (or
[bobenko_discrete_1996-1, Theorem 6]) that $f$ is isothermic if and only if
$\omega$ is closed, i.e. on every elementary quadrilateral $(ijk\ell)$,

$\omega_{ij}+\omega_{jk}+\omega_{k\ell}+\omega_{\ell i}=0.$

In this case, the dual surface $f^{*}$ of $f$, also called the _Christoffel
transform_ , can be recursively defined via

$\operatorname{d\\!}{f}_{ij}^{*}=\omega_{ij},$ (2.2)

and is also isothermic with identical cross-ratios factorising function
([bobenko_discrete_1996-1, p. 197])

$\operatorname{cr}(f^{*}_{i},f^{*}_{j},f^{*}_{k},f^{*}_{\ell})=\frac{\mu_{i\ell}}{\mu_{ij}}.$
(2.3)

We note here that the Christoffel transform of a discrete isothermic surface
is well-defined up to translations and homotheties. This freedom can be
observed in the choice of the cross-ratios factorising function $\mu$, as it
can be globally scaled by a constant.

Now to introduce the vector bundle approach to discrete isothermic surfaces in
the realm of Möbius geometry, take the trivial bundles
$\underline{\mathbb{H}}^{2}\mathrel{\mathop{\ordinarycolon}}=\Sigma^{2}\times\mathbb{H}^{2}$
and
$\underline{\mathbb{HP}}^{1}\mathrel{\mathop{\ordinarycolon}}=\Sigma^{2}\times\mathbb{HP}^{1}$
where $\mathbb{H}^{2}$ is considered as a right quaternionic vector space.
Then the points $f\in\mathbb{H}$ can be identified with its lift
$F\in\mathbb{HP}^{1}$ via

$\mathbb{H}\ni f\sim\begin{pmatrix}f\\\
1\end{pmatrix}\mathbb{H}=\mathrel{\mathop{\ordinarycolon}}\psi\mathbb{H}=\mathrel{\mathop{\ordinarycolon}}F\in\mathbb{HP}^{1}.$

With this identification, the lift
$F\mathrel{\mathop{\ordinarycolon}}\Sigma^{2}\to\mathbb{HP}^{1}$ of a discrete
surface $f$ can be viewed as a quaternionic line subbundle of the trivial
bundle $\underline{\mathbb{H}}^{2}$.

Now, for
$D(\lambda)_{ji}\mathrel{\mathop{\ordinarycolon}}\\{i\\}\times\mathbb{H}^{2}\to\\{j\\}\times\mathbb{H}^{2}$
with $\lambda\in\mathbb{R}$ defined on every edge $(ij)$ via

$D(\lambda)_{ji}\mathrel{\mathop{\ordinarycolon}}=\operatorname{id}_{ji}+\begin{pmatrix}0&\operatorname{d\\!}{f}_{ij}\\\
\lambda\omega_{ij}&0\end{pmatrix},$ (2.4)

let
$r(\lambda)_{ji}\mathrel{\mathop{\ordinarycolon}}\\{i\\}\times\mathbb{H}^{2}\to\\{j\\}\times\mathbb{H}^{2}$
be defined as

$r(\lambda)_{ji}\mathrel{\mathop{\ordinarycolon}}=(G\bullet
D(\lambda))_{ji}=G_{j}D(\lambda)_{ji}G_{i}^{-1}$ (2.5)

where $G=(e\,\,\psi)$ for $e=\begin{pmatrix}1\\\ 0\end{pmatrix}$. Then it is
known that $r(\lambda)$ can be written as

$r(\lambda)_{ji}=\operatorname{id}_{ji}+\lambda\eta_{ji}\mathrel{\mathop{\ordinarycolon}}=\operatorname{id}_{ji}+\lambda\begin{pmatrix}f_{j}\omega_{ij}&-f_{j}\omega_{ij}f_{i}\\\
\omega_{ij}&-\omega_{ij}f_{i}\end{pmatrix}.$ (2.6)

The projective transformation
$r(\lambda)^{P}_{ji}\mathrel{\mathop{\ordinarycolon}}\\{i\\}\times\mathbb{HP}^{1}\to\\{j\\}\times\mathbb{HP}^{1}$
induced by $r(\lambda)_{ji}$ defines a connection on the trivial bundle
$\underline{\mathbb{HP}}^{1}$. For an isothermic surface, connections in this
$1$-parameter family of connections are flat:

######  Fact 2.3 ([hertrich-jeromin_transformations_2000, Equation (13)],
[cho_periodic_nodate]).

Let $f\mathrel{\mathop{\ordinarycolon}}\Sigma^{2}\to\mathbb{H}$ be a discrete
surface, with its lift
$F\mathrel{\mathop{\ordinarycolon}}\Sigma^{2}\to\mathbb{HP}^{1}$. Then $f$ is
a discrete isothermic surface
$f\mathrel{\mathop{\ordinarycolon}}\Sigma^{2}\to\mathbb{H}$ if and only if
$r(\lambda)^{P}$ is flat. Moreover, if $\mu$ is the cross-ratios factorising
function of a discrete isothermic surface $f$, then $r(\lambda)$ can be
written as

$r(\lambda)_{ji}=\pi_{i}+\frac{\mu_{ij}-\lambda}{\mu_{ij}}\pi_{j}$

on every edge $(ij)$ where $\pi_{i}$ denotes the projection onto $F_{i}$.

######  Remark 2.4.

The $\eta_{ji}$ is called the _retraction form_ in [hertrich-
jeromin_transformations_2000, Definition 3.6].

####  2.2.2. Darboux transformations

On the other hand, the definition of Darboux transforms $\hat{f}$ of $f$ with
spectral parameter $\nu$ is given in [hertrich-jeromin_discrete_1999] via a
cross-ratios condition:

######  Definition 2.5.

Let $f\mathrel{\mathop{\ordinarycolon}}\Sigma^{2}\to\mathbb{H}$ be a discrete
isothermic surface with cross-ratios factorising function $\mu$. A second
discrete surface
$\hat{f}\mathrel{\mathop{\ordinarycolon}}\Sigma^{2}\to\mathbb{H}$ is called a
_Darboux transform of $f$ with spectral parameter $\nu$_ if

$\operatorname{cr}(f_{i},f_{j},\hat{f}_{j},\hat{f}_{i})=\frac{\nu}{\mu_{ij}}$

on every edge $(ij)$. The two surfaces $f$ and $\hat{f}$ are called a _Darboux
pair_ , and share the same cross-ratios factorising function.

A characterisation of Darboux transforms in terms of flat connections is given
in [hertrich-jeromin_transformations_2000, Lemma 3.17]:

######  Fact 2.6.

For two isothermic surfaces
$f,\hat{f}\mathrel{\mathop{\ordinarycolon}}\Sigma^{2}\to\mathbb{H}$, let
$F,\hat{F}\mathrel{\mathop{\ordinarycolon}}(\Sigma,\frac{1}{\mu})\to\mathbb{HP}^{1}$
be the corresponding lifts. Denoting by $r(\lambda)^{P}$, the associated
family of flat connections of $F$, we have that $\hat{F}$ is a Darboux
transform of $F$ with spectral parameter $\mu$ if and only if $\hat{F}$ is
$r(\nu)^{P}$–parallel, that is,

$r(\nu)^{P}_{ji}\hat{F}_{i}=\hat{F}_{j}$ (2.7)

on every edge $(ij)$.

We remark here that the Darboux transforms of a discrete isothermic surface
are uniquely determined by the choice of a spectral parameter and an initial
condition. Finally, we recall the condition for the Darboux transform
$\hat{f}$ of a discrete isothermic surface $f$ in $\mathbb{S}^{3}$ to take
values again in $\mathbb{S}^{3}$:

######  Fact 2.7 ([hertrich-jeromin_transformations_2000, p. 479], (cf.
[cho_periodic_nodate, Lemma 3.8])).

Let $f\mathrel{\mathop{\ordinarycolon}}\Sigma^{2}\to\mathbb{S}^{3}$ be a
discrete isothermic surface with its Darboux transform $\hat{f}$. Then
$\hat{f}$ takes values in $\mathbb{S}^{3}$ if and only if
$\hat{f}_{m_{0},n_{0}}\in\mathbb{S}^{3}$ for some
$(m_{0},n_{0})\in\Sigma^{2}$.

####  2.2.3. Constant mean curvature surfaces

An important subclass of isothermic surfaces is the class of constant mean
curvature (cmc) surfaces in $\mathbb{R}^{3}$. Discrete isothermic cmc surfaces
have been defined using various characterisations in the smooth case. Among
many, we highlight a few that are particularly relevant to us:

  * •

surfaces with constant mean curvature defined via Steiner’s formula
[schief_unification_2003, schief_maximum_2006, pottmann_geometry_2007,
bobenko_curvature_2010],

  * •

surfaces with Christoffel transform as parallel surfaces
[bobenko_discretization_1999, hertrich-jeromin_discrete_1999],

  * •

surfaces admitting a simultaneous Christoffel and Darboux transform [hertrich-
jeromin_discrete_1999],

  * •

surfaces admitting linear conserved quantities in the Minkowski $5$-space
[burstall_discrete_2014].

These various definitions have been shown to be equivalent (see, for example,
[bobenko_discrete_2008, burstall_discrete_2014]). For the purposes of the
examination of Darboux transforms of discrete cmc surfaces, we will use the
following definition:

######  Definition 2.8 ([hertrich-jeromin_discrete_1999, § 5]).

Let
$f\mathrel{\mathop{\ordinarycolon}}\Sigma^{2}\to\mathbb{R}^{3}\cong\operatorname{Im}\mathbb{H}$
be a discrete isothermic surface. Then $f$ is a discrete cmc $H$ surface if it
admits a Christoffel transform
$f^{*}\mathrel{\mathop{\ordinarycolon}}\Sigma^{2}\to\mathbb{R}^{3}$ so that

$|f^{*}_{i}-f_{i}|^{2}=\frac{1}{H^{2}}$

at every vertex $i\in\Sigma^{2}$. In this case, the Christoffel transform
$f^{*}$ is referred to as the parallel cmc surface of $f$.

This definition of discrete cmc surfaces via parallel cmc surfaces allows us
to recover another characterisation as follows:

######  Fact 2.9 ([hertrich-jeromin_discrete_1999, § 5]).

Let
$f\mathrel{\mathop{\ordinarycolon}}\Sigma^{2}\to\mathbb{R}^{3}\cong\operatorname{Im}\mathbb{H}$
be a discrete cmc surface with parallel cmc surface $f^{*}$. Then $f^{*}$ is a
Darboux transform of $f$ with spectral parameter $H^{2}$.

We have seen that discrete isothermic cmc surfaces admit a $4$-parameter
family of Darboux transforms that stay in $\mathbb{R}^{3}$. Subject to a
condition on the initial point, the Darboux transforms are again cmc surfaces:

######  Fact 2.10 ([hertrich-jeromin_discrete_1999, § 5]).

Let
$f\mathrel{\mathop{\ordinarycolon}}\Sigma^{2}\to\mathbb{R}^{3}\cong\operatorname{Im}\mathbb{H}$
be a discrete cmc-$H$ surface, and let
$\hat{f}\mathrel{\mathop{\ordinarycolon}}\Sigma^{2}\to\mathbb{R}^{3}\cong\operatorname{Im}\mathbb{H}$
be its Darboux transform with spectral parameter $\nu$. Then $\hat{f}$ is a
discrete cmc surface if and only if for some vertex $i\in\Sigma^{2}$, we have

$|\hat{f}_{i}-f^{*}_{i}|^{2}=\frac{1}{H^{2}}\left(1-\frac{H^{2}}{\nu}\right).$

######  Remark 2.11.

The condition for the Darboux transform of a cmc surfaces to be again a cmc
surface can also be given in terms of linear conserved quantities
[burstall_discrete_2014, Theorem 4.3].

###  2.3. Discrete polarised curves and the monodromy problem

We now briefly review the integrable theory of discrete polarised curves (see
[cho_periodic_nodate] for a detailed explanation). For some simply-connected
discrete domain $\Sigma^{1}\subset\mathbb{Z}$, let $\mu$ be a nowhere
vanishing function on the unoriented edges of $\Sigma^{1}$ so that
$(\Sigma^{1},\frac{1}{\mu})$ is a polarised domain. Discrete maps
$f\mathrel{\mathop{\ordinarycolon}}(\Sigma^{1},\frac{1}{\mu})\to\mathbb{H}$,
or their lifts
$F\mathrel{\mathop{\ordinarycolon}}(\Sigma^{1},\frac{1}{\mu})\to\mathbb{HP}^{1}$,
are called discrete polarised curves. Then the _dual curve_
$f^{*}\mathrel{\mathop{\ordinarycolon}}(\Sigma^{1},\frac{1}{\mu})\to\mathbb{H}$
of $f$ can be defined via

$\operatorname{d\\!}{f}^{*}_{ij}=\frac{1}{\mu_{ij}}\operatorname{d\\!}{f}_{ij}^{-1}.$

The integrable structure of discrete polarised curves can again be described
in terms of families of (flat) connections: putting
$D(\lambda)_{ji}\mathrel{\mathop{\ordinarycolon}}\\{i\\}\times\mathbb{H}^{2}\to\\{j\\}\times\mathbb{H}^{2}$
with $\lambda\in\mathbb{R}$ defined on every edge $(ij)$ via

$D(\lambda)_{ji}\mathrel{\mathop{\ordinarycolon}}=\operatorname{id}_{ji}+\begin{pmatrix}0&\operatorname{d\\!}{f}_{ij}\\\
\lambda\operatorname{d\\!}{f}^{*}_{ij}&0\end{pmatrix},$ (2.8)

let
$r(\lambda)_{ji}\mathrel{\mathop{\ordinarycolon}}\\{i\\}\times\mathbb{H}^{2}\to\\{j\\}\times\mathbb{H}^{2}$
be defined as

$r(\lambda)_{ji}\mathrel{\mathop{\ordinarycolon}}=(G\bullet
D(\lambda))_{ji}=G_{j}D(\lambda)_{ji}G_{i}$ (2.9)

where $G=(e\,\,\psi)$ so that

$\displaystyle r(\lambda)_{ji}$
$\displaystyle=\operatorname{id}_{ji}+\lambda\eta_{ji}\mathrel{\mathop{\ordinarycolon}}=\operatorname{id}_{ji}+\lambda\begin{pmatrix}f_{j}\operatorname{d\\!}{f}^{*}_{ij}&-f_{j}\operatorname{d\\!}{f}^{*}_{ij}f_{i}\\\
\operatorname{d\\!}{f}^{*}_{ij}&-\operatorname{d\\!}{f}^{*}_{ij}f_{i}\end{pmatrix}$
(2.10) $\displaystyle=\pi_{i}+\frac{\mu_{ij}-\lambda}{\mu_{ij}}\pi_{j}.$

The projective transformation
$r(\lambda)^{P}_{ji}\mathrel{\mathop{\ordinarycolon}}\\{i\\}\times\mathbb{HP}^{1}\to\\{j\\}\times\mathbb{HP}^{1}$
induced by $r(\lambda)_{ji}$ defines a connection on the trivial bundle
$\underline{\mathbb{HP}}^{1}$, called the _(flat) connection associated to
$f$_.

Darboux transformations of discrete polarised curves are given in terms of a
cross-ratios condition [cho_periodic_nodate, Definition 3.3] (see also
[cho_discrete_2021-1, Definition 3.1]):

######  Definition 2.12.

Two discrete polarised curves
$f,\hat{f}\mathrel{\mathop{\ordinarycolon}}(\Sigma^{1},\frac{1}{\mu})\to\mathbb{H}$
are called a _Darboux pair_ with spectral parameter $\nu$ if

$\operatorname{cr}(f,f_{j},\hat{f}_{j},\hat{f}_{i})=\frac{\nu}{\mu_{ij}}$

on every edge $(ij)$. In such case, one curve is called a _Darboux transform_
of the other curve with spectral parameter $\mu$.

The Darboux transformations of discrete polarised curves can be given a
characterisation in terms of the associated (flat) connections:

######  Fact 2.13.

Let
$F\mathrel{\mathop{\ordinarycolon}}(\Sigma^{1},\frac{1}{\mu})\to\mathbb{HP}^{1}$
be a discrete polarised curve with associated connection $r(\lambda)^{P}$.
Then
$\hat{F}\mathrel{\mathop{\ordinarycolon}}(\Sigma^{1},\frac{1}{\mu})\to\mathbb{HP}^{1}$
is a Darboux transform of $L$ with spectral parameter $\nu$ if and only if
$\hat{F}$ is $r(\nu)^{P}$–parallel.

For explicit calculations, the $r(\nu)^{P}$–parallel condition (2.7) for
Darboux transformations can be reformulated as follows. Suppose that $\hat{F}$
is $r(\nu)^{P}$–parallel so that for some $\hat{\psi}\in\Gamma\hat{F}$, we
have

$r(\nu)_{ji}\hat{\psi}_{i}=\hat{\psi}_{j},$

on any edge $(ij)$. Now, if we define
$\phi\mathrel{\mathop{\ordinarycolon}}=G^{-1}\hat{\psi}$, then we have

$D(\nu)_{ji}\phi_{i}=G^{-1}_{j}r(\nu)_{ji}\hat{\psi}_{i}=\phi_{j}.$

Therefore, if $\phi=\mathrel{\mathop{\ordinarycolon}}\begin{pmatrix}a\\\
b\end{pmatrix}$ for some
$a,b\mathrel{\mathop{\ordinarycolon}}\Sigma^{1}\to\mathbb{H}$, then $a$ and
$b$ must satisfy

$\begin{pmatrix}\operatorname{d\\!}{a}_{ij}\\\
\operatorname{d\\!}{b}_{ij}\end{pmatrix}=-\begin{pmatrix}\operatorname{d\\!}{f}_{ij}b_{i}\\\
\nu\operatorname{d\\!}{f}^{*}_{ij}a_{i}\end{pmatrix},$ (2.11)

and the new Darboux transform $\hat{f}$ is then given by $\hat{f}=f+ab^{-1}$.

To review the monodromy problem for Darboux transformations of discrete
polarised curves [cho_periodic_nodate, § 3.3], suppose that
$F,\hat{F}\mathrel{\mathop{\ordinarycolon}}(\Sigma^{1},\frac{1}{\mu})\to\mathbb{HP}^{1}$
are a Darboux pair of curves, and further assume that $F$ is an $M$-periodic
curve, that is,

$F_{m}=F_{m+M}$

for all $m\in\Sigma^{1}$. If the _monodromy matrix_ $\mathcal{M}_{r,\nu}$ is
defined by

$\mathcal{M}_{r,\nu}\mathrel{\mathop{\ordinarycolon}}=\prod_{\iota=m}^{m+M-1}r(\nu)_{(\iota,\iota+1)},$

then the eigenvectors $\phi$ of $\mathcal{M}_{r,\nu}$ correspond to _parallel
sections with multipliers_ , that is, for
$\hat{F}\mathrel{\mathop{\ordinarycolon}}=\operatorname{span}_{\mathbb{H}}\\{\phi\\}$,
we have

$\hat{F}_{m}=\hat{F}_{m+M}$

where $\hat{F}$ is $r(\nu)^{P}$–parallel. A spectral parameter $\nu$ is called
a _resonance point_ if every parallel section is a parallel section with
multiplier. In other words, using a resonance point as a spectral parameter
always gives closed Darboux transforms, regardless of the choice of the
initial condition.

##  3\. Monodromy of discrete isothermic surfaces

The integrable structure of discrete isothermic surfaces and that of discrete
polarised curves closely resemble each other. In this section, we formalise
the resemblance and interpret the integrability of discrete isothermic
surfaces in terms of transformations of discrete polarised curves. The
reduction to curves will be crucial to examining the monodromy of Darboux
transformations of discrete isothermic surfaces. We note here that such
reduction for the semi-discrete case has been considered in [burstall_semi-
discrete_2016], where semi-discrete isothermic surfaces [muller_semi-
discrete_2013] are characterised as successive Darboux transforms of smooth
polarised curves.

###  3.1. Discrete isothermic surfaces in terms of discrete polarised curves

Since we are interested in discrete curvature lines, let
$\Sigma^{2}\subset\mathbb{Z}^{2}$ be a discrete rectangular domain, that is,
$\Sigma^{2}=\Sigma^{1}\times\tilde{\Sigma}^{1}$ for simply-connected domains
$\Sigma^{1},\tilde{\Sigma}^{1}\subset\mathbb{Z}$.

Suppose that $f\mathrel{\mathop{\ordinarycolon}}\Sigma^{2}\to\mathbb{H}$ is a
discrete isothermic surface with cross-ratios factorising function $\mu$.
Viewing $\mu$ as a function defined on the unoriented edges of $\Sigma^{2}$,
we can consider the pair $(\Sigma^{2},\frac{1}{\mu})$ as a _discrete polarised
domain_ , and view discrete isothermic surfaces as defined on such polarised
domains. Therefore, given a discrete isothermic surface on a polarised domain,
all of its Christoffel transforms and Darboux transforms are defined on the
same polarised domain.

Denote by $f^{0}\mathrel{\mathop{\ordinarycolon}}\Sigma^{1}\to\mathbb{H}$ any
discrete curvature line of $f$ defined on the corresponding simply-connected
domain $\Sigma^{1}$. Since the cross-ratios factorising function $\mu$ serves
as the polarisation of the domain $\Sigma^{2}$, we can consider the
restriction of $\mu$ to the unoriented edges of $\Sigma^{1}$; therefore, we
see that
$f^{0}\mathrel{\mathop{\ordinarycolon}}(\Sigma^{1},\frac{1}{\mu})\to\mathbb{H}$
is a discrete polarised curve. In particular, we have that $\mu$ is an edge-
labeling; thus every discrete curvature line in a family is defined on the
same polarised domain $(\Sigma^{1},\frac{1}{\mu})$.

Take any two neighbouring discrete curvature lines $f^{0},f^{1}$ of $f$:
without loss of generality, let $f^{0}=f_{m,n_{0}}$ and $f^{1}=f_{m,n_{0}+1}$
for some fixed $n_{0}$. Then since $\mu$ is an edge-labeling, we can denote
the common polarisation on the edges $((m,n_{0}),(m,n_{0}+1))$ for any $m$ as
$\mu_{(n_{0},n_{0}+1)}$. The definition of discrete isothermic surfaces in
Definition 2.1 tells us that on any edge $(m,m+1)$, we have

$\operatorname{cr}(f_{m,n_{0}},f_{m+1,n_{0}},f_{m+1,n_{0}+1},f_{m,n_{0}+1})=\frac{\mu_{(n_{0},n_{0}+1)}}{\mu_{(m,m+1)}}.$

Thus, the definition of Darboux transformations of discrete polarised curves
in Definition 2.12 allows us to deduce the following:

######  Proposition 3.1.

Any two neighbouring discrete curvature lines of a discrete isothermic surface
form a Darboux pair of discrete polarised curves.

######  Remark 3.2.

Note that the cross-ratios factorising functions are well-defined only up to
constant multiplication. Thus, when restricting the polarised domain of a
discrete isothermic surface to the polarised domain of a curvature line, there
is a freedom of choice in the scaling of the polarisation.

In a similar manner, for a Darboux pair of discrete isothermic surfaces
$f,\hat{f}\mathrel{\mathop{\ordinarycolon}}(\Sigma^{2},\frac{1}{\mu})\to\mathbb{H}$,
consider a corresponding pair of discrete curvature lines
$f^{0},\hat{f}^{0}\mathrel{\mathop{\ordinarycolon}}(\Sigma^{1},\frac{1}{\mu})\to\mathbb{H}$
defined on a restricted discrete polarised domain. Then we have for any edge
$(ij)\subset\Sigma^{1}$,

$\operatorname{cr}(f^{0}_{i},f^{0}_{j},\hat{f}^{0}_{j},\hat{f}^{0}_{i})=\frac{\nu}{\mu_{ij}}.$

Thus the definition of Darboux transformations of discrete isothermic surfaces
in Definition 2.5 directly yield the following observation:

######  Proposition 3.3.

Any two corresponding discrete curvature lines of a discrete Darboux pair of
isothermic surfaces form a Darboux pair of discrete polarised curves.

Furthermore, the well-definedness of Darboux transformations of discrete
isothermic surfaces implies the permutability of Darboux transformations of
discrete polarised curves:

######  Theorem 3.4.

Let
$f\mathrel{\mathop{\ordinarycolon}}(\Sigma^{1},\frac{1}{\mu})\to\mathbb{H}$ be
a polarised curve, with Darboux transforms
$f_{1},f_{2}\mathrel{\mathop{\ordinarycolon}}(\Sigma^{1},\frac{1}{\mu})\to\mathbb{H}$
of $f$ with respective spectral parameter $\nu_{1},\nu_{2}$. Then there exists
a fourth discrete polarised curve
$f_{12}\mathrel{\mathop{\ordinarycolon}}(\Sigma^{1},\frac{1}{\mu})\to\mathbb{H}$
that is a simultaneous Darboux transform of $f_{1}$ and $f_{2}$ with
respective spectral parameter $\nu_{2}$ and $\nu_{1}$.

###  3.2. Reduction of the monodromy problem

So far, we have obtained the reduction of the integrable structure of discrete
isothermic surfaces and its Darboux transformations into a lattice of Darboux
transformations of discrete polarised curves. Such change in perspective is
crucial for our consideration of the monodromy problem of Darboux transforms
of discrete isothermic surfaces.

Thus, let $f$ be a discrete isothermic surface with a period $M$; without loss
of generality, we will assume that $f$ is periodic in the $m$-curvature line
direction so that $f_{m,n}=f_{m+M,n}$ for some $M\in\mathbb{N}$.

Now suppose that $\hat{f}$ is a Darboux transformation of $f$ with spectral
parameter $\nu$. Let us further assume that one of the $m$-curvature lines of
$\hat{f}$ is periodic with the same period as $f$, i.e.

$\hat{f}_{m,n_{0}}=\hat{f}_{m+M,n_{0}}$

for some fixed $n_{0}$. Then from the definition of Darboux transformations,
the cross-ratios condition

$\operatorname{cr}(f_{m,n_{0}},f_{m,n_{0}+1},\hat{f}_{m,n_{0}+1},\hat{f}_{m,n_{0}})=\frac{\nu}{\mu_{(n_{0},n_{0}+1)}}$

for any $m$ implies that $\hat{f}_{m,n_{0}+1}$ must be determined uniquely
from the values of

$f_{m,n_{0}},\quad
f_{m,n_{0}+1},\quad\hat{f}_{m,n_{0}},\quad\nu,\quad\mu_{(n_{0},n_{0}+1)}.$

However, all of these values are $M$-periodic in the $m$-direction; thus, we
have that $\hat{f}_{m,n_{0}+1}$ must also be periodic, that is,

$\hat{f}_{m,n_{0}+1}=\hat{f}_{m+M,n_{0}+1}.$

Propogating to the entire domain, we conclude that the entire surface is
periodic, i.e., for any $n$,

$\hat{f}_{m,n}=\hat{f}_{m+M,n}.$

We have thus proved that the monodromy problem of Darboux transformations of
discrete isothermic surfaces can be reduced to a monodromy problem of Darboux
transformations of a single discrete curvature line:

######  Theorem 3.5.

Let
$f\mathrel{\mathop{\ordinarycolon}}(\Sigma^{2},\frac{1}{\mu})\to\mathbb{H}$ be
a discrete isothermic surface with one family of periodic curvature lines. If
$\hat{f}\mathrel{\mathop{\ordinarycolon}}(\Sigma^{2},\frac{1}{\mu})\to\mathbb{H}$
is a Darboux transform of $f$, then the corresponding family of curvature
lines of $\hat{f}$ are also periodic with the same period if and only if one
of the corresponding family of curvature lines is periodic.

Therefore, if $f$ is a discrete isothermic surfaces with periodic
$m$-curvature lines, we can obtain new periodic isothermic surfaces as
follows:

  1. (1)

Choose any base point $f_{m_{0},n_{0}}$ of $f$ and the $m$-curvature line
$f_{m,n_{0}}$ through that point.

  2. (2)

Viewing $f_{m,n_{0}}$ as a closed discrete polarised curve, find the spectral
parameter $\nu$ and the initial condition $\hat{f}_{m_{0},n_{0}}$ to obtain
closed Darboux transform $\hat{f}_{m,n_{0}}$ (as a polarised curve).

  3. (3)

Use the spectral parameter $\nu$ and the initial condition
$\hat{f}_{m_{0},n_{0}}$ to obtain a new periodic discrete isothermic surface
$\hat{f}$.

In particular, if we have that all curvature lines of a discrete isothermic
surface are periodic so that $f$ is a discrete torus, then we can also obtain
new discrete isothermic tori as follows:

  1. (1)

Choose any base point $f_{m_{0},n_{0}}$ of $f$, and take the $m$-curvature
line $f_{m,n_{0}}$ and $n$-curvature line $f_{m_{0},n}$ through that point.

  2. (2)

Viewing both curves as closed discrete polarised curves, find a spectral
parameter $\nu$ and an initial condition $\hat{f}_{m_{0},n_{0}}$ to obtain
simultaneously closed Darboux transforms $\hat{f}_{m,n_{0}}$ and
$\hat{f}_{m_{0},n}$ of $f_{m,n_{0}}$ and $f_{m_{0},n}$, respectively.

  3. (3)

Use the spectral parameter $\nu$ and the initial condition
$\hat{f}_{m_{0},n_{0}}$ to obtain a new periodic discrete isothermic torus
$\hat{f}$.

The main obstacle for the construction of new discrete isothermic tori via
Darboux transformations is the second step.

##  4\. Periodic discrete isothermic surfaces

In this section, we put our theory to test to obtain new examples of periodic
discrete isothermic surfaces. In particular, we will consider the Darboux
transforms of surfaces of revolution in $\mathbb{R}^{3}$, and obtain discrete
analogues of isothermic bubbletons (see, for example, [cho_generalised_2022, §
2.2]) via Darboux transformations. By restricting to the case of discrete
circular cylinders, we will also obtain the discrete analogues of cmc
cylinders called bubbletons [sterling_existence_1993]. Finally, considering
the Darboux transforms of homogeneous tori in $\mathbb{S}^{3}$, we will
construct the discrete analogues of isothermic tori found in [bernstein_non-
special_2001] (see also [hertrich-jeromin_introduction_2003, Example 5.4.25]).

###  4.1. Closed Darboux transforms of discrete circles – revisited

The commonground for all the cases we consider in the section is the fact that
the closed curvature lines will be multiply-covered discrete circles with
constant polarisation. Thus we first review the closure conditions for the
Darboux transforms of multiply-covered discrete circles, where we generalise
the calculations from [cho_periodic_nodate, Example 3.8] to allow for any
radius and any constant polarisation. We note here that we will only consider
the discrete circles constructed via uniform sampling of the smooth circle, to
obtain explicit results.

Let
$f\mathrel{\mathop{\ordinarycolon}}(\Sigma^{1},\frac{1}{\mu})\to\mathbb{R}^{2}\subset\mathbb{R}^{4}\cong\operatorname{Im}\mathbb{H}$
be a $\rho$–fold cover of the discrete circle given by

$f_{m}\mathrel{\mathop{\ordinarycolon}}=\mathbbm{j}re^{\frac{2\pi\mathbbm{i}}{M}m}$

for some $r\in\mathbb{R}_{+}$ and $M\in\mathbb{N}$. Throughout the paper, we
will assume that $M>2$. Give $f$ a constant polarisation which we write as

$\frac{1}{\mu_{ij}}=\alpha|1-e^{\frac{2\pi\mathbbm{i}}{M}}|^{2}$

for some non-zero real constant $\alpha$. Note that for $\alpha=r^{2}$, then
we have

$\frac{1}{\mu_{ij}}=r^{2}|1-e^{\frac{2\pi\mathbbm{i}}{M}}|^{2}=|f_{i}-f_{j}|^{2}=|{\operatorname{d\\!}{f}_{ij}}|^{2}$

so that $f$ is polarised by arc-length. Then we obtain that $f$ is an $\rho
M$–periodic discrete polarised curve for any $\rho\in\mathbb{N}$.

To consider the Darboux transforms, let $\hat{F}=\hat{\psi}\mathbb{H}$ be
$r(\nu)^{P}$–parallel, and $\phi=\begin{pmatrix}a\\\ b\end{pmatrix}$ where
$\phi=G^{-1}\hat{\psi}$. Rewriting (2.11), we have that on any three
consecutive vertices $(ijp)$, $a$ must satisfy the recurrence relation

$\nu\operatorname{d\\!}{f}_{ij}^{*}a_{i}-(\operatorname{d\\!}{f}_{ij})^{-1}\operatorname{d\\!}{a}_{ij}+(\operatorname{d\\!}{f}_{jp})^{-1}\operatorname{d\\!}{a}_{jp}=0.$

In the case of discrete circles under consideration, the recurrence relation
reads

$e^{\frac{2\pi\mathbbm{i}}{M}}a_{p}-(1+e^{\frac{2\pi\mathbbm{i}}{M}})a_{j}+(1-\nu\alpha|1-e^{\frac{2\pi\mathbbm{i}}{M}}|^{2})a_{i}=0.$

If we set $a=\mathrel{\mathop{\ordinarycolon}}a_{0}+\mathbbm{j}a_{1}$, for
some functions
$a_{0},a_{1}\mathrel{\mathop{\ordinarycolon}}\Sigma^{2}\to\mathbb{C}$, we can
solve the recurrence equation and obtain

$a=a_{0}^{+}c_{0}+a_{0}^{-}c_{1}+\mathbbm{j}(a_{1}^{+}c_{2}+a_{1}^{-}c_{3})$
(4.1)

for some constants $c_{0},c_{1},c_{2},c_{3}\in\mathbb{C}$, and

$\displaystyle a_{0,m}^{\pm}$
$\displaystyle=\left(\tfrac{1}{2}(e^{-\frac{2\pi\mathbbm{i}}{M}}(1\pm s)+(1\mp
s))\right)^{m}$ (4.2) $\displaystyle a_{1,m}^{\pm}$
$\displaystyle=\left(\tfrac{1}{2}(e^{\frac{2\pi\mathbbm{i}}{M}}(1\mp s)+(1\pm
s))\right)^{m}$

where $s\mathrel{\mathop{\ordinarycolon}}=\sqrt{1-4\nu\alpha}$.

To consider the monodromy, we notice that $\phi=\begin{pmatrix}a\\\
b\end{pmatrix}$ is an eigenvector of the monodromy matrix if and only if

$\phi_{m}h=\phi_{m+\rho M}$ (4.3)

for some $h\in\mathbb{H}$. However, due to the recurrence relation for $a,b$
in (2.11), we have that (4.3) holds if and only if

$a_{m}h=a_{m+\rho M}.$

To investigate when the curve admits resonance points, we define
$h_{\iota}^{\pm}$ so that

$a^{\pm}_{\iota,m}h_{\iota}^{\pm}=a^{\pm}_{\iota,m+\rho M}$

for $\iota=0,1$, and look for conditions under which

$h_{0}^{+}=h_{0}^{-}=h_{1}^{+}=h_{1}^{-}.$ (4.4)

Since $a^{\pm}_{\iota,m+\rho M}=a^{\pm}_{\iota,m}a^{\pm}_{\iota,\rho M}$, we
have $h_{\iota}^{\pm}=a^{\pm}_{\iota,\rho M}.$ However, note that

$\displaystyle h_{0}^{\pm}=a^{\pm}_{0,\rho M}$
$\displaystyle=\left(\tfrac{1}{2}(e^{-\frac{2\pi\mathbbm{i}}{M}}(1\pm s)+(1\mp
s))\right)^{\rho M}$
$\displaystyle=\left(\tfrac{1}{2}e^{-\frac{2\pi\mathbbm{i}}{M}}((1\pm
s)+e^{\frac{2\pi\mathbbm{i}}{M}}(1\mp s))\right)^{\rho M}$
$\displaystyle=a^{\pm}_{1,\rho M}=h_{1}^{\pm}.$

Therefore, we have (4.4) if and only if $h_{1}^{+}=h_{1}^{-}$, that is,

$e^{2\pi\mathbbm{i}k}=\frac{h_{1}^{+}}{h_{1}^{-}}=\frac{a^{+}_{1,\rho
M}}{a^{-}_{1,\rho
M}}=\left(\frac{e^{\frac{2\pi\mathbbm{i}}{M}}(1-s)+(1+s)}{e^{\frac{2\pi\mathbbm{i}}{M}}(1+s)+(1-s)}\right)^{\rho
M}$

for some $k\in\mathbb{Z}$. Thus, we have that $\nu$ is a resonance point of
$f$ if and only if

$\nu=\frac{1}{4\alpha}\left(1-\cot^{2}\frac{\pi}{M}\tan^{2}\frac{k\pi}{\rho
M}\right).$ (4.5)

We will assume without loss of generality that $\rho,k\in\mathbb{N}$.

We will now obtain the explicit parametrisations of closed Darboux transforms
$\hat{f}$ of the discrete circle. Taking a resonance point (4.5) as the
spectral parameter, we have $s\in\mathbb{R}$ so that
$a_{0}^{\pm}=\overline{a_{1}^{\mp}}$. Thus, $a$ can be written as

$\displaystyle a$
$\displaystyle=a_{0}^{+}c_{0}+a_{0}^{-}c_{1}+\overline{a_{1}^{+}}\mathbbm{j}c_{2}+\overline{a_{1}^{-}}\mathbbm{j}c_{3}=a_{0}^{+}(c_{0}+\mathbbm{j}c_{3})+a_{0}^{-}(c_{1}+\mathbbm{j}c_{2})$
$\displaystyle=\mathrel{\mathop{\ordinarycolon}}a_{0}^{+}c^{+}+a_{0}^{-}c^{-},$

for some constants $c^{\pm}\in\mathbb{H}$. We can further calculate that

$a^{\pm}_{0,m}=\left(e^{-\frac{\pi\mathbbm{i}}{M}}\cos\tfrac{\pi}{M}\sec\tfrac{k\pi}{\rho
M}\right)^{m}e^{\mp\frac{k\pi\mathbbm{i}}{\rho M}m}$

so that $a$ admits the following explicit form:

$a_{m}=\left(e^{-\frac{\pi\mathbbm{i}}{M}}\cos\tfrac{\pi}{M}\sec\tfrac{k\pi}{\rho
M}\right)^{m}\left(e^{-\frac{k\pi\mathbbm{i}}{\rho
M}m}c^{+}+e^{\frac{k\pi\mathbbm{i}}{\rho M}m}c^{-}\right).$

By (2.11), we have that

$\displaystyle b_{m}$
$\displaystyle=-\operatorname{d\\!}{f}_{ij}^{-1}\operatorname{d\\!}{a}_{ij}$
$\displaystyle=\frac{\left(e^{-\frac{\pi\mathbbm{i}}{M}}\cos\tfrac{\pi}{M}\sec\tfrac{k\pi}{\rho
M}\right)^{m}\mathbbm{j}\left(\sin{\frac{(\rho+k)\pi}{\rho
M}}e^{-\frac{k\pi\mathbbm{i}}{\rho M}m}c^{+}+\sin{\frac{(\rho-k)\pi}{\rho
M}}e^{\frac{k\pi\mathbbm{i}}{\rho
M}m}c^{-}\right)}{2r\sin{\tfrac{\pi}{M}}\cos{\tfrac{k\pi}{\rho M}}}.$

Thus, every Darboux transform of a discrete circle $f$ is given by

$\displaystyle\hat{f}_{m}$ $\displaystyle=f_{m}+a_{m}b_{m}^{-1}$
$\displaystyle=\mathbbm{j}re^{\frac{2\pi\mathbbm{i}}{M}m}+2r\sin{\tfrac{\pi}{M}}\cos{\tfrac{k\pi}{\rho
M}}e^{-\frac{\pi\mathbbm{i}}{M}m}\alpha_{m}\beta_{m}^{-1}e^{\frac{\pi\mathbbm{i}}{M}m}$

where

$\displaystyle\alpha_{m}$ $\displaystyle=e^{-\frac{k\pi\mathbbm{i}}{\rho
M}m}c^{+}+e^{\frac{k\pi\mathbbm{i}}{\rho M}m}c^{-},$ $\displaystyle\beta_{m}$
$\displaystyle=\mathbbm{j}\left(e^{-\frac{k\pi\mathbbm{i}}{\rho
M}m}\sin{\tfrac{(\rho+k)\pi}{\rho M}}c^{+}+e^{\frac{k\pi\mathbbm{i}}{\rho
M}m}\sin{\tfrac{(\rho-k)\pi}{\rho M}}c^{-}\right)$

for some constants $c^{\pm}\in\mathbb{H}$.

To find those Darboux transforms $\hat{f}$ living in the same
$\mathbbm{j}\mathbbm{k}$–plane as $f$, we can assume without loss of
generality that $c^{+}=\mathbbm{j}$ and $c^{-}=\mathbbm{j}c_{2}$ for some
$c_{2}\in\mathbb{C}$. Thus we obtain the explicit parametrisation of closed
Darboux transforms of discrete circles living in the same
$\mathbbm{j}\mathbbm{k}$–plane:

$\hat{f}_{m}=-\mathbbm{j}re^{\frac{2\pi\mathbbm{i}}{M}m}\,\frac{e^{-\frac{k\pi\mathbbm{i}}{\rho
M}m}\sin\frac{(\rho+k)\pi}{\rho M}c_{2}+e^{\frac{k\pi\mathbbm{i}}{\rho
M}m}\sin\frac{(\rho-k)\pi}{\rho M}}{e^{-\frac{k\pi\mathbbm{i}}{\rho
M}m}\sin\frac{(\rho-k)\pi}{\rho M}c_{2}+e^{\frac{k\pi\mathbbm{i}}{\rho
M}m}\sin\frac{(\rho+k)\pi}{\rho M}}.$ (4.6)

###  4.2. Discrete isothermic cylinders

Suppose we have a discrete surface of revolution in
$\mathbb{R}^{3}\cong\operatorname{Im}\mathbb{H}$ given by

$f_{m,n}\mathrel{\mathop{\ordinarycolon}}=\mathbbm{i}q_{n}+\mathbbm{j}p_{n}e^{\frac{2\pi\mathbbm{i}}{M}m}$

for some functions
$p,q\mathrel{\mathop{\ordinarycolon}}\Sigma^{1}\to\mathbb{R}$. As calculated
in [burstall_discrete_2014, § 2], we then have that for $i=(m,n)$,

$\operatorname{cr}(f_{i},f_{j},f_{k},f_{\ell})=-\frac{p_{n}p_{n+1}}{(p_{n}-p_{n+1})^{2}+(q_{n}-q_{n+1})^{2}}|1-e^{\frac{2\pi\mathbbm{i}}{M}}|^{2}=\frac{\mu_{i\ell}}{\mu_{ij}}.$
(4.7)

Clearly, $f$ has $M$-periodic $m$-curvature lines.

To consider the monodromy of its Darboux transforms over $\rho$–fold cover in
the $m$-direction, we use Theorem 3.5 and consider a single $m$-curvature
line, say $f_{m,n_{0}}$ for some fixed $n_{0}$. By Remark 3.2, we have some
freedom to choose the correct polarisation $\mu$ for the discrete polarised
curve $f_{m,n_{0}}$ from (4.7). Thus we choose

$\frac{1}{\mu_{ij}}=|1-e^{\frac{2\pi\mathbbm{i}}{M}}|^{2}$

so that any two neighbouring $m$-curvature lines will be a Darboux pair with
spectral parameter $\mu_{i\ell}$ where

$\frac{1}{\mu_{i\ell}}=-\frac{(p_{n}-p_{n+1})^{2}+(q_{n}-q_{n+1})^{2}}{p_{n}p_{n+1}}.$

If we assume without loss of generality that $q_{n_{0}}=0$, then we have

$f_{m,n_{0}}=\mathbbm{j}p_{n_{0}}e^{\frac{2\pi\mathbbm{i}}{M}m}.$

Thus we have that $f_{m,n_{0}}$ is a discrete polarised circle with constant
polarisation.

Using the discussions in Section 4.1, if we choose the spectral parameter
$\nu$ as the resonance point (4.5) with $\alpha=1$, then any Darboux transform
$\hat{f}_{m,n_{0}}$ of the discrete polarised curve $f_{m,n_{0}}$ will be
periodic. Therefore, Theorem 3.5 tells us that any Darboux transform $\hat{f}$
of $f$ with spectral parameter $\nu$ satisfying

$\nu=\frac{1}{4}\left(1-\cot^{2}\frac{\pi}{M}\tan^{2}\frac{k\pi}{\rho
M}\right)$

gives a new discrete isothermic cylinder with period $\rho M$.

###  4.3. Discrete isothermic bubbletons via discrete circular cylinder

Now suppose that $f$ is a discrete circular cylinder, given by

$f_{m,n}\mathrel{\mathop{\ordinarycolon}}=\mathbbm{i}q_{n}+\mathbbm{j}e^{\frac{2\pi\mathbbm{i}}{M}m}.$

Then this is a special case of the discrete surface of revolution, so that

$\operatorname{cr}(f_{i},f_{j},f_{k},f_{\ell})=-\frac{1}{(q_{n}-q_{n+1})^{2}}|1-e^{\frac{2\pi\mathbbm{i}}{M}}|^{2}=\frac{\mu_{i\ell}}{\mu_{ij}}.$

We again take

$\frac{1}{\mu_{ij}}=|1-e^{\frac{2\pi\mathbbm{i}}{M}}|^{2},\quad\frac{1}{\mu_{i\ell}}=-(q_{n}-q_{n+1})^{2}.$

Then any Darboux transform $\hat{f}$ of $f$ with spectral parameter

$\nu=\frac{1}{4}\left(1-\cot^{2}\frac{\pi}{M}\tan^{2}\frac{k\pi}{\rho
M}\right)$ (4.8)

gives a discrete isothermic cylinder.

We now aim to obtain explicit parametrisations for such discrete cylinders:
for this, we assume that

  1. (1)

the spectral parameter $\nu$ is a resonance point (4.8), and

  2. (2)

$q_{n}=\frac{n}{N}$ for some $N\in\mathbb{Z}$.

We will solve for
$a,b\mathrel{\mathop{\ordinarycolon}}\Sigma^{2}\to\mathbb{H}$ so that
$\hat{f}=f+ab^{-1}$, that is, $a$ and $b$ satisfies the linearised Riccati
equation (2.11) on any edge $(ij)$. Since we have
$f_{m,0}=\mathbbm{j}e^{\frac{2\pi\mathbbm{i}}{M}m}$, the discussions in
Section 4.1 tells us that

$\displaystyle a_{m,0}$
$\displaystyle=\left(e^{-\frac{\pi\mathbbm{i}}{M}}\cos\tfrac{\pi}{M}\sec\tfrac{k\pi}{\rho
M}\right)^{m}\alpha_{m},$ (4.9) $\displaystyle b_{m,0}$
$\displaystyle=\frac{1}{2r}\csc{\tfrac{\pi}{M}}\sec{\tfrac{k\pi}{\rho
M}}\left(e^{-\frac{\pi\mathbbm{i}}{M}}\cos\tfrac{\pi}{M}\sec\tfrac{k\pi}{\rho
M}\right)^{m}\beta_{m}$

where

$\displaystyle\alpha_{m}$ $\displaystyle=e^{-\frac{k\pi\mathbbm{i}}{\rho
M}m}c^{+}+e^{\frac{k\pi\mathbbm{i}}{\rho M}m}c^{-},$ $\displaystyle\beta_{m}$
$\displaystyle=\mathbbm{j}\left(e^{-\frac{k\pi\mathbbm{i}}{\rho
M}m}\sin{\tfrac{(\rho+k)\pi}{\rho M}}c^{+}+e^{\frac{k\pi\mathbbm{i}}{\rho
M}m}\sin{\tfrac{(\rho-k)\pi}{\rho M}}c^{-}\right)$

for some $c^{\pm}\in\mathbb{H}$.

Now, fixing any $m=m_{0}\in\mathbb{Z}$, we will solve for $a,b$ over the line
$(m_{0},n)$. Thus, we first need to solve for $a$ where $a$ satisfies the
recurrence relation

$\nu\operatorname{d\\!}{f}_{i\ell}^{*}a_{i}-(\operatorname{d\\!}{f}_{i\ell})^{-1}\operatorname{d\\!}{a}_{i\ell}+(\operatorname{d\\!}{f}_{\ell
q})^{-1}\operatorname{d\\!}{a}_{\ell q}=0$

over any three consecutive vertices
$i=(m_{0},n),\ell=(m_{0},n+1),q=(m_{0},n+2)$. Under current assumptions, the
recurrence relation reads

$a_{q}-2a_{\ell}+\left(1+\frac{\nu}{N^{2}}\right)a_{i}=0.$

This allows us to solve for $a_{m_{0},n}$ explicitly, and obtain that

$a_{m,n}=\left(1+\tfrac{\sqrt{-\nu}}{N}\right)^{n}\gamma_{m}^{+}+\left(1-\tfrac{\sqrt{-\nu}}{N}\right)^{n}\gamma_{m}^{-}$

where $\gamma^{\pm}$ are quaternionic-valued functions over $m$ alone. Then
the explicit expression for $b$ follows via (2.11):

$b_{m,n}=\mathbbm{i}\left(\left(1+\tfrac{\sqrt{-\nu}}{N}\right)^{n}\gamma_{m}^{+}-\left(1-\tfrac{\sqrt{-\nu}}{N}\right)^{n}\gamma_{m}^{-}\right)\sqrt{-\nu}.$

To find $\gamma_{m}^{\pm}$, note that when $n=0$, we have

$a_{m,0}=\gamma_{m}^{+}+\gamma_{m}^{-}\quad\text{and}\quad
b_{m,0}=\mathbbm{i}(\gamma_{m}^{+}-\gamma_{m}^{-})\sqrt{-\nu}$

so that

$\gamma_{m}^{\pm}=\frac{1}{2}\left(a_{m,0}\mp\mathbbm{i}b_{m,0}\frac{1}{\sqrt{-\nu}}\right).$

Using (4.9), we find that

$\gamma_{m}^{\pm}=\frac{1}{2}\left(e^{-\frac{\pi\mathbbm{i}}{M}}\cos\tfrac{\pi}{M}\sec\tfrac{k\pi}{\rho
M}\right)^{m}\left(\alpha_{m}\mp\frac{\mathbbm{i}}{2\sqrt{-\nu}}\csc{\tfrac{\pi}{M}}\sec{\tfrac{k\pi}{\rho
M}}\beta_{m}\right).$

Thus, we have found closed-form solutions for the functions $a$ and $b$, which
in turn yields the explicit parametrisations of the discrete isothermic
cylinder that are Darboux transforms of the discrete circular cylinder via

$\hat{f}_{m,n}=f_{m,n}+a_{m,n}b_{m,n}^{-1}.$

######  Example 4.1.

For example, if we additionally assume that

  1. (1)

the spectral parameter at the resonance point satisfies $\nu<0$, and

  2. (2)

$\hat{f}_{m,0}$ lives in the same $\mathbbm{j}\mathbbm{k}$–plane as
$f_{m,0}=\mathbbm{j}e^{\frac{2\pi\mathbbm{i}}{M}m}$,

then without loss of generality we may set $c^{+}=\mathbbm{j}$ and
$c^{-}=\mathbbm{j}c_{2}$ for some $c_{2}\in\mathbb{C}$. Then we have

$\displaystyle\alpha_{m}$
$\displaystyle=\mathbbm{j}\left(e^{\frac{k\pi\mathbbm{i}}{\rho
M}m}+e^{-\frac{k\pi\mathbbm{i}}{\rho
M}m}c_{2}\right)=\mathrel{\mathop{\ordinarycolon}}\mathbbm{j}A_{m}$
$\displaystyle\beta_{m}$ $\displaystyle=-\left(e^{\frac{k\pi\mathbbm{i}}{\rho
M}m}\sin{\tfrac{(\rho+k)\pi}{\rho M}}+e^{-\frac{k\pi\mathbbm{i}}{\rho
M}m}\sin{\tfrac{(\rho-k)\pi}{\rho
M}}c_{2}\right)=\mathrel{\mathop{\ordinarycolon}}-B_{m},$

so that $\gamma^{\pm}$ takes the following form:

$\gamma_{m}^{\pm}=\frac{1}{2}\left(e^{-\frac{\pi\mathbbm{i}}{M}}\cos\tfrac{\pi}{M}\sec\tfrac{k\pi}{\rho
M}\right)^{m}\left(\mathbbm{j}A_{m}\pm\frac{\mathbbm{i}}{2\sqrt{-\nu}}\csc{\tfrac{\pi}{M}}\sec{\tfrac{k\pi}{\rho
M}}B_{m}\right).$

From this, we calculate that

$\displaystyle|\gamma_{m}^{\pm}|^{2}$
$\displaystyle=\frac{1}{4}\left(\cos\tfrac{\pi}{M}\sec\tfrac{k\pi}{\rho
M}\right)^{2m}\left(|A_{m}|^{2}-\tfrac{1}{4\nu}\csc^{2}{\tfrac{\pi}{M}}\sec^{2}{\tfrac{k\pi}{\rho
M}}|B_{m}|^{2}\right),$
$\displaystyle\gamma_{m}^{+}\overline{\gamma_{m}^{-}}+\gamma_{m}^{-}\overline{\gamma_{m}^{+}}$
$\displaystyle=\frac{1}{2}\left(\cos\tfrac{\pi}{M}\sec\tfrac{k\pi}{\rho
M}\right)^{2m}\left(|A_{m}|^{2}+\tfrac{1}{4\nu}\csc^{2}{\tfrac{\pi}{M}}\sec^{2}{\tfrac{k\pi}{\rho
M}}|B_{m}|^{2}\right),$
$\displaystyle\gamma_{m}^{-}\overline{\gamma_{m}^{+}}-\gamma_{m}^{+}\overline{\gamma_{m}^{-}}$
$\displaystyle=-\mathbbm{j}\frac{\mathbbm{i}}{2\sqrt{-\nu}}\left(e^{\frac{\pi\mathbbm{i}}{M}}\cos\tfrac{\pi}{M}\sec\tfrac{k\pi}{\rho
M}\right)^{2m}\csc{\tfrac{\pi}{M}}\sec{\tfrac{k\pi}{\rho
M}}A_{m}\overline{B_{m}}.$

Using these equations, we can write the following expressions explicitly:

$\displaystyle|b_{m,n}|^{2}$
$\displaystyle=\begin{multlined}-\nu\Big{(}\big{(}1+\tfrac{\sqrt{-\nu}}{N}\big{)}^{2n}|\gamma_{m}^{+}|^{2}+\big{(}1-\tfrac{\sqrt{-\nu}}{N}\big{)}^{2n}|\gamma_{m}^{-}|^{2}\\\
\phantom{aaaaa}-\big{(}1+\tfrac{\sqrt{-\nu}}{N}\big{)}^{n}\big{(}1-\tfrac{\sqrt{-\nu}}{N}\big{)}^{n}(\gamma_{m}^{+}\overline{\gamma_{m}^{-}}+\gamma_{m}^{-}\overline{\gamma_{m}^{+}})\Big{)},\end{multlined}-\nu\Big{(}\big{(}1+\tfrac{\sqrt{-\nu}}{N}\big{)}^{2n}|\gamma_{m}^{+}|^{2}+\big{(}1-\tfrac{\sqrt{-\nu}}{N}\big{)}^{2n}|\gamma_{m}^{-}|^{2}\\\
\phantom{aaaaa}-\big{(}1+\tfrac{\sqrt{-\nu}}{N}\big{)}^{n}\big{(}1-\tfrac{\sqrt{-\nu}}{N}\big{)}^{n}(\gamma_{m}^{+}\overline{\gamma_{m}^{-}}+\gamma_{m}^{-}\overline{\gamma_{m}^{+}})\Big{)},$
$\displaystyle a_{m,n}\overline{b_{m,n}}$
$\displaystyle=\begin{multlined}-\sqrt{-\nu}\Big{(}\big{(}1+\tfrac{\sqrt{-\nu}}{N}\big{)}^{2n}|\gamma_{m}^{+}|^{2}-\big{(}1-\tfrac{\sqrt{-\nu}}{N}\big{)}^{2n}|\gamma_{m}^{-}|^{2}\\\
\phantom{aaaaa}+\big{(}1+\tfrac{\sqrt{-\nu}}{N}\big{)}^{n}\big{(}1-\tfrac{\sqrt{-\nu}}{N}\big{)}^{n}(\gamma_{m}^{-}\overline{\gamma_{m}^{+}}-\gamma_{m}^{+}\overline{\gamma_{m}^{-}})\Big{)}\mathbbm{i}.\end{multlined}-\sqrt{-\nu}\Big{(}\big{(}1+\tfrac{\sqrt{-\nu}}{N}\big{)}^{2n}|\gamma_{m}^{+}|^{2}-\big{(}1-\tfrac{\sqrt{-\nu}}{N}\big{)}^{2n}|\gamma_{m}^{-}|^{2}\\\
\phantom{aaaaa}+\big{(}1+\tfrac{\sqrt{-\nu}}{N}\big{)}^{n}\big{(}1-\tfrac{\sqrt{-\nu}}{N}\big{)}^{n}(\gamma_{m}^{-}\overline{\gamma_{m}^{+}}-\gamma_{m}^{+}\overline{\gamma_{m}^{-}})\Big{)}\mathbbm{i}.$

Therefore, we obtain the closed-form parametrisation of the discrete
isothermic cylinders $\hat{f}$ via

$\hat{f}=f+ab^{-1}=f+a\bar{b}|b|^{-2}=\mathrel{\mathop{\ordinarycolon}}f+\frac{1}{C}(T^{0}+\mathbbm{j}T^{1})$
(4.10)

where

$\displaystyle C_{m,n}$
$\displaystyle=\begin{multlined}2\big{(}(N-\sqrt{-\nu})^{n}+(N+\sqrt{-\nu})^{n}\big{)}^{2}|B_{m}|^{2}\\\
+\big{(}(N-\sqrt{-\nu})^{n}-(N+\sqrt{-\nu})^{n}\big{)}^{2}|A_{m}|^{2}(\cos\tfrac{2\pi}{M}-\cos\tfrac{2k\pi}{\rho
M})\end{multlined}2\big{(}(N-\sqrt{-\nu})^{n}+(N+\sqrt{-\nu})^{n}\big{)}^{2}|B_{m}|^{2}\\\
+\big{(}(N-\sqrt{-\nu})^{n}-(N+\sqrt{-\nu})^{n}\big{)}^{2}|A_{m}|^{2}(\cos\tfrac{2\pi}{M}-\cos\tfrac{2k\pi}{\rho
M})$ $\displaystyle T^{0}_{m,n}$
$\displaystyle=\frac{\mathbbm{i}}{\sqrt{-\nu}}\big{(}(N-\sqrt{-\nu})^{2n}-(N+\sqrt{-\nu})^{2n}\big{)}\big{(}2|B_{m}|^{2}+|A_{m}|^{2}(\cos\tfrac{2\pi}{M}-\cos\tfrac{2k\pi}{\rho
M})\big{)}$ $\displaystyle T^{1}_{m,n}$
$\displaystyle=-16e^{\frac{2\pi\mathbbm{i}}{M}m}(N-\sqrt{-\nu})^{n}(N+\sqrt{-\nu})^{n}\sin\tfrac{\pi}{M}\cos\tfrac{k\pi}{\rho
M}A_{m}\overline{B_{m}}.$

See Figures 3 and 4 for examples.

Figure 3. Discrete isothermic bubbletons drawn using explicit parametrisation
(4.10) with $k=2$, $\rho=1$, and $c_{2}=-10$ with various number of
subdivisions (on the left: $M=40$, $N=5$, on the right: $M=80$, $N=10$).

Figure 4. Discrete isothermic bubbletons drawn using explicit parametrisation
(4.10) with $k=3$, $\rho=1$, $M=80$, and $N=8$ with varying initial conditions
(on the left: $c_{2}=-4$, on the right: $c_{2}=-8$).

###  4.4. Discrete cmc bubbletons

Discrete circular cylinders are discrete cmc surfaces, which can be verified
as follows: noting that

$\displaystyle f^{*}_{m,n}-f^{*}_{m+1,n}$
$\displaystyle=\operatorname{d\\!}{f}^{*}_{ij}=\frac{1}{\mu_{ij}}\operatorname{d\\!}{f}_{ij}^{-1}=-\mathbbm{j}e^{\frac{2\pi\mathbbm{i}}{M}m}+\mathbbm{j}e^{\frac{2\pi\mathbbm{i}}{M}(m+1)}$
$\displaystyle f^{*}_{m,n}-f^{*}_{m,n+1}$
$\displaystyle=\operatorname{d\\!}{f}^{*}_{i\ell}=\frac{1}{\mu_{i\ell}}\operatorname{d\\!}{f}_{i\ell}^{-1}=\mathbbm{i}q_{n}-\mathbbm{i}q_{n+1},$

we take

$f^{*}_{0,0}=\mathbbm{i}q_{0}-\mathbbm{j}$

as the initial condition to obtain

$f^{*}_{m,n}=\mathbbm{i}q_{n}-\mathbbm{j}e^{\frac{2\pi\mathbbm{i}}{M}m}$

as the dual surface. Therefore, we have

$f^{*}_{m,n}-f_{m,n}=-2\mathbbm{j}e^{\frac{2\pi\mathbbm{i}}{M}m},$

so that

$|f^{*}-f|^{2}=4,$

allowing us to conclude that $f$ is a discrete cmc surface with
$H\equiv\frac{1}{2}$ by Definition 2.8.

Thus, if we can choose the correct initial condition using Fact 2.10, then we
will obtain new discrete cmc cylinders via Darboux transformations. To find
such initial condition, we first note that

$\operatorname{cr}(f_{i},f_{j},f^{*}_{j},f^{*}_{i})=\frac{1}{4}|1-e^{\frac{2\pi\mathbbm{i}}{M}}|^{2}\quad\text{and}\quad\operatorname{cr}(f_{i},f_{\ell},f^{*}_{\ell},f^{*}_{i})=-\frac{1}{4}(q_{n}-q_{n+1})^{2}$

so that $f^{*}$ is a Darboux transform of $f$ with spectral parameter
$H^{2}=\frac{1}{4}$ (confirming Fact 2.9). Thus for some point
$i\in\Sigma^{2}$, we must find $\hat{f}_{i}$ so that

$|\hat{f}_{i}-f^{*}_{i}|^{2}=4\left(1-\frac{1}{4\nu}\right),$

which exists if and only if $\nu>\frac{1}{4}$ or $\nu<0$.

Figure 5. Discrete cmc bubbletons drawn using explicit parametrisation (4.10)
where $k=2$, $\rho=1$ with initial condition as in (4.11) at various number of
subdivisions (on the top left: $M=40$, $N=5$; on the top right: $M=40$,
$N=10$; on the bottom left: $M=160$, $N=5$; on the bottom right: $M=160$,
$N=15$).

To investigate the conditions on $k$ and $\rho$ for the spectral parameter at
a resonance point

$\nu=\frac{1}{4}\left(1-\cot^{2}\frac{\pi}{M}\tan^{2}\frac{k\pi}{\rho
M}\right)$

to satisfy either $\nu>\frac{1}{4}$ or $\nu<0$, first note that $k=\rho$
implies $\nu=0$; thus, we must have $k\neq\rho$. Suppose that $0<k<\rho$. On
the inverval $u\in(0,\frac{\pi}{2})$, $\sin u$ is a strictly increasing
positive function while $\cos u$ is a strictly decreasing positive function so
that

$0<\sin\frac{k\pi}{\rho M}<\sin\frac{\pi}{M}\quad\text{and}\quad
0<\cos\frac{\pi}{M}<\cos\frac{k\pi}{\rho M}.$

Thus, we deduce

$0<\tan\frac{k\pi}{\rho M}<\tan\frac{\pi}{M},$

that is,

$0<\cot^{2}\frac{\pi}{M}\tan^{2}\frac{k\pi}{\rho M}<1.$

This implies that

$0<\nu<\frac{1}{4}$

so that we do not have discrete cmc cylinders appearing as Darboux transforms
of the discrete circular cylinders when $k<\rho$. This is the discrete
analogue of the smooth case investigated in [sterling_existence_1993] (see
also [cho_new_2022]).

On the other hand, if $k>\rho$, then for large enough $M$ so that
$\frac{k}{\rho M}<\frac{1}{2}$, we have

$0<\sin\frac{\pi}{M}<\sin\frac{k\pi}{\rho M}<1\quad\text{and}\quad
0<\cos\frac{k\pi}{\rho M}<\cos\frac{\pi}{M}.$

Thus, in this case

$0<\tan\frac{\pi}{M}<\tan\frac{k\pi}{\rho M}$

so that

$1<\cot^{2}\frac{\pi}{M}\tan^{2}\frac{k\pi}{\rho M}.$

In other words, we have $\nu<0$. Thus, when we choose the spectral parameter
$\nu$ at the resonance point with $k>\rho$ with large enough $M$, we can find
a correct initial condition to obtain another discrete cmc cylinder via
Darboux transformation. These are the discrete analogues of the bubbletons
constructed in [sterling_existence_1993].

Figure 6. Discrete cmc bubbletons drawn using explicit parametrisation (4.10)
with initial condition as in (4.11) (on the left: $k=3$, $\rho=1$, $M=120$,
$N=15$; on the right: $k=4$, $\rho=3$, $M=80$, $N=10$).

######  Remark 4.2.

The condition that $M$ is sufficiently large is essential, setting the
discrete case apart from the smooth case. In the smooth case, the condition
$k>\rho$ is sufficient to guarantee the existence of cmc cylinders appearing
as Darboux transforms of the circular cylinder; however in the discrete case,
there are configurations where $k>\rho$ is not sufficient. For example, if we
choose $k=5$, $\rho=2$, and $M=3$, then we have

$0<\nu=\frac{1}{4}\left(1-\cot^{2}\frac{\pi}{3}\tan^{2}\frac{5\pi}{6}\right)=\frac{2}{9}<\frac{1}{4}.$

To obtain closed-form parametrisations of discrete cmc bubbletons, the fact
that $\nu<0$ allows us to use Example 4.1; therefore, let $\hat{f}$ be
parametrised by (4.10) so that via (4.6), we have

$\hat{f}_{0,0}=-\mathbbm{j}\,\frac{\sin\frac{(\rho+k)\pi}{\rho
M}c_{2}+\sin\frac{(\rho-k)\pi}{\rho M}}{\sin\frac{(\rho-k)\pi}{\rho
M}c_{2}+\sin\frac{(\rho+k)\pi}{\rho M}}.$

For simplicity, we will assume that $c_{2}\in\mathbb{R}$ so that
$\hat{f}_{0,0}=\mathbbm{j}q$ for some $q\in\mathbb{R}$. Thus, we need to find
$c_{2}$ such that

$4-\frac{1}{\nu}=|\hat{f}_{0,0}-f^{*}_{0,0}|^{2}=\frac{4(c_{2}-1)^{2}\cos^{2}\frac{\pi}{M}\sin^{2}\frac{k\pi}{\rho
M}}{\big{(}\sin\frac{(\rho-k)\pi}{\rho M}c_{2}+\sin\frac{(\rho+k)\pi}{\rho
M}\big{)}^{2}},$

and a direct calculation shows

$c_{2}=\pm\frac{\sqrt{\cos\frac{2\pi}{M}-\cos\frac{2k\pi}{\rho
M}}}{\sqrt{2}\sin\frac{(\rho-k)\pi}{\rho M}}.$ (4.11)

For such $c_{2}$, we can use the closed-form parametrisation given in (4.10)
to obtain discrete cmc bubbletons. See Figure 5, 6, and 7 for examples.

Figure 7. Discrete cmc bubbletons drawn using explicit parametrisation (4.10)
where $k=5$, $M=40$, and $N=5$ with initial condition as in (4.11) at varying
number of covers (top row from the left: $\rho=1,2$; bottom row from the left:
$\rho=3,4$).

###  4.5. Discrete isothermic tori

To construct non-trivial examples of discrete isothermic tori, let us consider
the discrete homogeneous tori in $\mathbb{S}^{3}$ as our given surface $f$ so
that

$f_{m,n}\mathrel{\mathop{\ordinarycolon}}=qe^{\frac{2\pi\mathbbm{i}}{N}n}+\mathbbm{j}pe^{\frac{2\pi\mathbbm{i}}{M}m}$

for non-zero $M,N\in\mathbb{Z}$ and $p,q\in\mathbb{R}$ satisfying
$p^{2}+q^{2}=1$. We take the $\rho_{1}$–fold cover in the $m$-direction and
$\rho_{2}$–fold cover in the $n$-direction. Then we calculate that

$\operatorname{cr}(f_{i},f_{j},f_{k},f_{\ell})=-\frac{p^{2}|1-e^{\frac{2\pi\mathbbm{i}}{M}}|^{2}}{q^{2}|1-e^{\frac{2\pi\mathbbm{i}}{N}}|^{2}}=\frac{\mu_{i\ell}}{\mu_{ij}}.$

Thus we assume without loss of generality that

$\frac{1}{\mu_{ij}}=p^{2}|1-e^{\frac{2\pi\mathbbm{i}}{M}}|^{2},\quad\frac{1}{\mu_{i\ell}}=-q^{2}|1-e^{\frac{2\pi\mathbbm{i}}{N}}|^{2},$

so that every $m$-curvature line is arc-length polarised while every
$n$-curvature line is negative arc-length polarised.

Choosing any base point $(m_{0},n_{0})$, let us first take the $m$-curvature
line $f_{m,n_{0}}$ through the base point. Then since we have a discrete
circle with constant polarisation, we deduce that any of its Darboux
transforms $\hat{f}_{m,n_{0}}$ with spectral parameter

$\nu_{1}=\frac{1}{4p^{2}}\left(1-\cot^{2}\frac{\pi}{M}\tan^{2}\frac{k_{1}\pi}{\rho_{1}M}\right)$

for any $k_{1}\in\mathbb{Z}$ results in a closed discrete polarised curve.
Similarly, the $n$-curvature line $f_{m_{0},n}$ is also a discrete circle with
constant polarisation, so that any of its Darboux transforms
$\hat{f}_{m_{0},n}$ with spectral parameter

$\nu_{2}=-\frac{1}{4q^{2}}\left(1-\cot^{2}\frac{\pi}{N}\tan^{2}\frac{k_{2}\pi}{\rho_{2}N}\right)$

for $k_{2}\in\mathbb{Z}$ results in a closed discrete polarised curve.

Thus, if we choose $p$ and $q$ so that

$\displaystyle p^{2}$
$\displaystyle=\frac{1-\cot^{2}\frac{\pi}{M}\tan^{2}\frac{k_{1}\pi}{\rho_{1}M}}{\cot^{2}\frac{\pi}{N}\tan^{2}\frac{k_{2}\pi}{\rho_{2}N}-\cot^{2}\frac{\pi}{M}\tan^{2}\frac{k_{1}\pi}{\rho_{1}M}},$
(4.12) $\displaystyle q^{2}$
$\displaystyle=\frac{1-\cot^{2}\frac{\pi}{N}\tan^{2}\frac{k_{2}\pi}{\rho_{2}N}}{\cot^{2}\frac{\pi}{M}\tan^{2}\frac{k_{1}\pi}{\rho_{1}M}-\cot^{2}\frac{\pi}{N}\tan^{2}\frac{k_{2}\pi}{\rho_{2}N}},$

then we have $p^{2}+q^{2}=1$, and

$\nu\mathrel{\mathop{\ordinarycolon}}=\nu_{1}=\nu_{2}=\frac{1}{4}\left(\cot^{2}\frac{\pi}{N}\tan^{2}\frac{k_{2}\pi}{\rho_{2}M}-\cot^{2}\frac{\pi}{M}\tan^{2}\frac{k_{1}\pi}{\rho_{1}M}\right).$
(4.13)

This implies that for such $p$ and $q$, we have that every Darboux transform
with such spectral parameter $\nu=\nu_{1}=\nu_{2}$ gives a discrete isothermic
torus. These are the discrete analogues of the isothermic tori found in
[bernstein_non-special_2001] (see also [hertrich-jeromin_introduction_2003]).

To obtain closed-form parametrisations of the discrete isothermic tori, let us
take $p$ and $q$ as in (4.12) so that the spectral parameter satisfies

$\nu=\frac{1}{4p^{2}}\left(1-\cot^{2}\frac{\pi}{M}\tan^{2}\frac{k_{1}\pi}{\rho_{1}M}\right)=-\frac{1}{4q^{2}}\left(1-\cot^{2}\frac{\pi}{N}\tan^{2}\frac{k_{2}\pi}{\rho_{2}N}\right).$

Since $f_{m,0}=q+\mathbbm{j}pe^{\frac{2\pi\mathbbm{i}}{M}}m$, we use the
discussions from Section 4.1 to obtain that

$\displaystyle a_{m,0}$
$\displaystyle=\left(e^{-\frac{\pi\mathbbm{i}}{M}}\cos\tfrac{\pi}{M}\sec\tfrac{k_{1}\pi}{\rho_{1}M}\right)^{m}\alpha_{m},$
(4.14) $\displaystyle b_{m,0}$
$\displaystyle=\tfrac{1}{2p}\csc{\tfrac{\pi}{M}}\sec{\tfrac{k_{1}\pi}{\rho_{1}M}}\left(e^{-\frac{\pi\mathbbm{i}}{M}}\cos\tfrac{\pi}{M}\sec\tfrac{k_{1}\pi}{\rho_{1}M}\right)^{m}\beta_{m}$

where

$\displaystyle\alpha_{m}$
$\displaystyle=e^{-\frac{k_{1}\pi\mathbbm{i}}{\rho_{1}M}m}c^{+}+e^{\frac{k_{1}\pi\mathbbm{i}}{\rho_{1}M}m}c^{-},$
$\displaystyle\beta_{m}$
$\displaystyle=\mathbbm{j}\left(e^{-\frac{k_{1}\pi\mathbbm{i}}{\rho_{1}M}m}\sin{\tfrac{(\rho_{1}+k_{1})\pi}{\rho_{1}M}}c^{+}+e^{\frac{k_{1}\pi_{1}\mathbbm{i}}{\rho_{1}M}m}\sin{\tfrac{(\rho_{1}-k_{1})\pi}{\rho_{1}M}}c^{-}\right)$

for some $c^{\pm}\in\mathbb{H}$.

Now to solve along an $n$-curvature line direction, we fix some $m$ and note
that $n$-curvature lines are also circles; thus, we can use a completely
analogous argument in Section 4.1 to find that

$\displaystyle a_{m,n}$
$\displaystyle=\big{(}e^{\frac{\pi\mathbbm{i}}{N}}\cos\tfrac{\pi}{N}\sec\tfrac{k_{2}\pi}{\rho_{2}N}\big{)}^{n}\Big{(}e^{\frac{k_{2}\pi\mathbbm{i}}{\rho_{2}N}n}\gamma^{+}_{m}+e^{-\frac{k_{2}\pi\mathbbm{i}}{\rho_{2}N}n}\gamma^{-}_{m}\Big{)},$
(4.15) $\displaystyle b_{m,n}$
$\displaystyle=\begin{multlined}-\tfrac{1}{2q}\csc{\tfrac{\pi}{N}}\sec{\tfrac{k_{2}\pi}{\rho_{2}N}}\big{(}e^{-\frac{\pi\mathbbm{i}}{N}}\cos\tfrac{\pi}{N}\sec\tfrac{k_{2}\pi}{\rho_{2}N}\big{)}^{n}\\\
\qquad\Big{(}e^{\frac{k_{2}\pi\mathbbm{i}}{\rho_{2}N}n}\sin\tfrac{(\rho_{2}+k_{2})\pi}{\rho_{2}N}\gamma^{+}_{m}+e^{-\frac{k_{2}\pi\mathbbm{i}}{\rho_{2}N}n}\sin\tfrac{(\rho_{2}-k_{2})\pi}{\rho_{2}N}\gamma^{-}_{m}\Big{)}\end{multlined}-\tfrac{1}{2q}\csc{\tfrac{\pi}{N}}\sec{\tfrac{k_{2}\pi}{\rho_{2}N}}\big{(}e^{-\frac{\pi\mathbbm{i}}{N}}\cos\tfrac{\pi}{N}\sec\tfrac{k_{2}\pi}{\rho_{2}N}\big{)}^{n}\\\
\qquad\Big{(}e^{\frac{k_{2}\pi\mathbbm{i}}{\rho_{2}N}n}\sin\tfrac{(\rho_{2}+k_{2})\pi}{\rho_{2}N}\gamma^{+}_{m}+e^{-\frac{k_{2}\pi\mathbbm{i}}{\rho_{2}N}n}\sin\tfrac{(\rho_{2}-k_{2})\pi}{\rho_{2}N}\gamma^{-}_{m}\Big{)}$

for some quaternion valued functions $\gamma^{\pm}_{m}$ of $m$. To find
$\gamma^{\pm}_{m}$, we let $n=0$ in (4.15) to see that

$\displaystyle a_{m,0}$ $\displaystyle=\gamma^{+}_{m}+\gamma^{-}_{m},$
$\displaystyle b_{m,0}$
$\displaystyle=-\tfrac{1}{2q}\csc{\tfrac{\pi}{N}}\sec{\tfrac{k_{2}\pi}{\rho_{2}N}}\Big{(}\sin\tfrac{(\rho_{2}+k_{2})\pi}{\rho_{2}N}\gamma^{+}_{m}+\sin\tfrac{(\rho_{2}-k_{2})\pi}{\rho_{2}N}\gamma^{-}_{m}\Big{)}.$

Solving for $\gamma^{\pm}_{m}$ and using (4.14), we obtain that

$\displaystyle\gamma^{\pm}_{m}$
$\displaystyle=\\!\begin{multlined}\mp\tfrac{1}{2}\sec{\tfrac{\pi}{N}}\csc{\tfrac{k_{2}\pi}{\rho_{2}N}}\big{(}e^{-\frac{\pi\mathbbm{i}}{M}}\cos\tfrac{\pi}{M}\sec\tfrac{k_{1}\pi}{\rho_{1}M}\big{)}^{m}\\\
\qquad\Big{(}\sin\tfrac{(\rho_{2}\mp
k_{2})\pi}{\rho_{2}N}\alpha_{m}+\tfrac{q}{p}\sin{\tfrac{\pi}{N}}\cos{\tfrac{k_{2}\pi}{\rho_{2}N}}\csc{\tfrac{\pi}{M}}\sec{\tfrac{k_{1}\pi}{\rho_{1}N}}\beta_{m}\Big{)}\end{multlined}\mp\tfrac{1}{2}\sec{\tfrac{\pi}{N}}\csc{\tfrac{k_{2}\pi}{\rho_{2}N}}\big{(}e^{-\frac{\pi\mathbbm{i}}{M}}\cos\tfrac{\pi}{M}\sec\tfrac{k_{1}\pi}{\rho_{1}M}\big{)}^{m}\\\
\qquad\Big{(}\sin\tfrac{(\rho_{2}\mp
k_{2})\pi}{\rho_{2}N}\alpha_{m}+\tfrac{q}{p}\sin{\tfrac{\pi}{N}}\cos{\tfrac{k_{2}\pi}{\rho_{2}N}}\csc{\tfrac{\pi}{M}}\sec{\tfrac{k_{1}\pi}{\rho_{1}N}}\beta_{m}\Big{)}$
$\displaystyle=\mathrel{\mathop{\ordinarycolon}}\mp\tfrac{1}{2}\sec{\tfrac{\pi}{N}}\csc{\tfrac{k_{2}\pi}{\rho_{2}N}}\big{(}e^{-\frac{\pi\mathbbm{i}}{M}}\cos\tfrac{\pi}{M}\sec\tfrac{k_{1}\pi}{\rho_{1}M}\big{)}^{m}\tilde{\gamma}^{\pm}_{m}.$

Therefore,

$\displaystyle a_{m,n}$
$\displaystyle=\\!\begin{multlined}-\tfrac{1}{2}\sec{\tfrac{\pi}{N}}\csc{\tfrac{k_{2}\pi}{\rho_{2}N}}\big{(}e^{-\frac{\pi\mathbbm{i}}{M}}\cos\tfrac{\pi}{M}\sec\tfrac{k_{1}\pi}{\rho_{1}M}\big{)}^{m}\big{(}e^{\frac{\pi\mathbbm{i}}{N}}\cos\tfrac{\pi}{N}\sec\tfrac{k_{2}\pi}{\rho_{2}N}\big{)}^{n}\\\
\qquad\Big{(}e^{\frac{k_{2}\pi\mathbbm{i}}{\rho_{2}N}n}\tilde{\gamma}^{+}_{m}-e^{-\frac{k_{2}\pi\mathbbm{i}}{\rho_{2}N}n}\tilde{\gamma}^{-}_{m}\Big{)}\end{multlined}-\tfrac{1}{2}\sec{\tfrac{\pi}{N}}\csc{\tfrac{k_{2}\pi}{\rho_{2}N}}\big{(}e^{-\frac{\pi\mathbbm{i}}{M}}\cos\tfrac{\pi}{M}\sec\tfrac{k_{1}\pi}{\rho_{1}M}\big{)}^{m}\big{(}e^{\frac{\pi\mathbbm{i}}{N}}\cos\tfrac{\pi}{N}\sec\tfrac{k_{2}\pi}{\rho_{2}N}\big{)}^{n}\\\
\qquad\Big{(}e^{\frac{k_{2}\pi\mathbbm{i}}{\rho_{2}N}n}\tilde{\gamma}^{+}_{m}-e^{-\frac{k_{2}\pi\mathbbm{i}}{\rho_{2}N}n}\tilde{\gamma}^{-}_{m}\Big{)}$
$\displaystyle b_{m,n}$
$\displaystyle=\\!\begin{multlined}\tfrac{1}{q}\csc\tfrac{2\pi}{N}\csc\tfrac{2k_{2}\pi}{\rho_{2}N}\big{(}e^{-\frac{\pi\mathbbm{i}}{M}}\cos\tfrac{\pi}{M}\sec\tfrac{k_{1}\pi}{\rho_{1}M}\big{)}^{m}\big{(}e^{-\frac{\pi\mathbbm{i}}{N}}\cos\tfrac{\pi}{N}\sec\tfrac{k_{2}\pi}{\rho_{2}N}\big{)}^{n}\\\
\qquad\Big{(}e^{\frac{k_{2}\pi\mathbbm{i}}{\rho_{2}N}n}\sin\tfrac{(\rho_{2}+k_{2})\pi}{\rho_{2}N}\tilde{\gamma}^{+}_{m}-e^{-\frac{k_{2}\pi\mathbbm{i}}{\rho_{2}N}n}\sin\tfrac{(\rho_{2}-k_{2})\pi}{\rho_{2}N}\tilde{\gamma}^{-}_{m}\Big{)}.\end{multlined}\tfrac{1}{q}\csc\tfrac{2\pi}{N}\csc\tfrac{2k_{2}\pi}{\rho_{2}N}\big{(}e^{-\frac{\pi\mathbbm{i}}{M}}\cos\tfrac{\pi}{M}\sec\tfrac{k_{1}\pi}{\rho_{1}M}\big{)}^{m}\big{(}e^{-\frac{\pi\mathbbm{i}}{N}}\cos\tfrac{\pi}{N}\sec\tfrac{k_{2}\pi}{\rho_{2}N}\big{)}^{n}\\\
\qquad\Big{(}e^{\frac{k_{2}\pi\mathbbm{i}}{\rho_{2}N}n}\sin\tfrac{(\rho_{2}+k_{2})\pi}{\rho_{2}N}\tilde{\gamma}^{+}_{m}-e^{-\frac{k_{2}\pi\mathbbm{i}}{\rho_{2}N}n}\sin\tfrac{(\rho_{2}-k_{2})\pi}{\rho_{2}N}\tilde{\gamma}^{-}_{m}\Big{)}.$

Summarzing, we have

$a_{m,n}b_{m,n}^{-1}=\\!\begin{multlined}-2q\sin\tfrac{\pi}{N}\cos\tfrac{k_{2}\pi}{\rho_{2}N}e^{-\frac{\pi\mathbbm{i}}{M}m}e^{\frac{\pi\mathbbm{i}}{N}n}\Big{(}e^{\frac{k_{2}\pi\mathbbm{i}}{\rho_{2}N}n}\tilde{\gamma}^{+}_{m}-e^{-\frac{k_{2}\pi\mathbbm{i}}{\rho_{2}N}n}\tilde{\gamma}^{-}_{m}\Big{)}\\\
\Big{(}e^{\frac{k_{2}\pi\mathbbm{i}}{\rho_{2}N}n}\sin\tfrac{(\rho_{2}+k_{2})\pi}{\rho_{2}N}\tilde{\gamma}^{+}_{m}-e^{-\frac{k_{2}\pi\mathbbm{i}}{\rho_{2}N}n}\sin\tfrac{(\rho_{2}-k_{2})\pi}{\rho_{2}N}\tilde{\gamma}^{-}_{m}\Big{)}^{-1}e^{\frac{\pi\mathbbm{i}}{M}m}e^{\frac{\pi\mathbbm{i}}{N}n},\end{multlined}-2q\sin\tfrac{\pi}{N}\cos\tfrac{k_{2}\pi}{\rho_{2}N}e^{-\frac{\pi\mathbbm{i}}{M}m}e^{\frac{\pi\mathbbm{i}}{N}n}\Big{(}e^{\frac{k_{2}\pi\mathbbm{i}}{\rho_{2}N}n}\tilde{\gamma}^{+}_{m}-e^{-\frac{k_{2}\pi\mathbbm{i}}{\rho_{2}N}n}\tilde{\gamma}^{-}_{m}\Big{)}\\\
\Big{(}e^{\frac{k_{2}\pi\mathbbm{i}}{\rho_{2}N}n}\sin\tfrac{(\rho_{2}+k_{2})\pi}{\rho_{2}N}\tilde{\gamma}^{+}_{m}-e^{-\frac{k_{2}\pi\mathbbm{i}}{\rho_{2}N}n}\sin\tfrac{(\rho_{2}-k_{2})\pi}{\rho_{2}N}\tilde{\gamma}^{-}_{m}\Big{)}^{-1}e^{\frac{\pi\mathbbm{i}}{M}m}e^{\frac{\pi\mathbbm{i}}{N}n},$
(4.16)

where

$\tilde{\gamma}^{\pm}_{m}=\sin\tfrac{(\rho_{2}\mp
k_{2})\pi}{\rho_{2}N}\alpha_{m}+\tfrac{q}{p}\sin{\tfrac{\pi}{N}}\cos{\tfrac{k_{2}\pi}{\rho_{2}N}}\csc{\tfrac{\pi}{M}}\sec{\tfrac{k_{1}\pi}{\rho_{1}M}}\beta_{m}$

for

$\displaystyle\alpha_{m}$
$\displaystyle=e^{-\frac{k_{1}\pi\mathbbm{i}}{\rho_{1}M}m}c^{+}+e^{\frac{k_{1}\pi\mathbbm{i}}{\rho_{1}M}m}c^{-},$
$\displaystyle\beta_{m}$
$\displaystyle=\mathbbm{j}\left(e^{-\frac{k_{1}\pi\mathbbm{i}}{\rho_{1}M}m}\sin{\tfrac{(\rho_{1}+k_{1})\pi}{\rho_{1}M}}c^{+}+e^{\frac{k_{1}\pi_{1}\mathbbm{i}}{\rho_{1}M}m}\sin{\tfrac{(\rho_{1}-k_{1})\pi}{\rho_{1}M}}c^{-}\right),$

and the discrete isothermic tori $\hat{f}$ obtained via Darboux
transformations of the homogeneous tori are given by

$\hat{f}_{m,n}=f_{m,n}+a_{m,n}b_{m,n}^{-1}.$

If the constants $c^{\pm}\in\mathbb{H}$ are chosen so that for any initial
point $(m_{0},n_{0})$, we have $\hat{f}_{m_{0},n_{0}}\in\mathbb{S}^{3}$, then
it follows that $\hat{f}_{m,n}$ is a discrete isothermic torus in
$\mathbb{S}^{3}$ (see Fact 2.7).

Figure 8. Discrete isothermic tori in $\mathbb{S}^{3}$ drawn using the
explicit parametrisation (4.16) with $k_{1}=4$, $\rho_{1}=3$, $k_{2}=2$,
$\rho_{2}=3$, $c^{+}=0.45+\mathbbm{j}r_{2}$, and $c^{-}=1$ with different
number of subdivisions and corresponding values of $r_{2}$ (on the top left:
$M=12$, $N=40$, $r_{2}\approx 0.395155$; on the top right: $M=40$, $N=40$,
$r_{2}\approx 0.385119$; on the bottom left: $M=160$, $N=40$, $r_{2}\approx
0.384212$; on the bottom right: $M=160$, $N=160$, $r_{2}\approx 0.384438$).

Figure 9. Discrete isothermic tori in $\mathbb{S}^{3}$ drawn using the
explicit parametrisation (4.16) (on the left: $k_{1}=2$, $\rho_{1}=3$,
$k_{2}=3$, $\rho_{2}=2$, $M=12$, $N=12$, $c^{+}\approx
0.45+\mathbbm{j}0.236786$, $c^{-}=1$; on the right: $k_{1}=3$, $\rho_{1}=2$,
$k_{2}=2$, $\rho_{2}=3$, $M=60$, $N=40$,
$c^{+}\approx-0.6-\mathbbm{j}1.70678$, $c^{-}=1$).

###  4.6. Continuum limit of the resonance points

Note that the calculation of resonance points in each section is dependent on
the number of subdivisions of the smooth circle. Therefore, for the resonance
points, taking the continuum limit corresponds to taking the limit as the
number of subdivisions tend to infinity. As we have investigated the monodromy
problem of the discrete isothermic surfaces under structure-preserving
discretisation, it is fair to expect that the resonance points of the discrete
objects also converge to those of the smooth counterpart under the continuum
limit. We can check the convergence explicitly as follows.

For the surface of revolution case, we see that

$\lim_{M\to\infty}\frac{1}{4}\left(1-\cot^{2}\frac{\pi}{M}\tan^{2}\frac{k\pi}{\rho
M}\right)=\frac{\rho^{2}-k^{2}}{4\rho^{2}}$

for $k\in\mathbb{Z}$, which is identical to the resonance point of surface of
revolutions in the smooth case (see, for example, [cho_generalised_2022,
Section 2]).

Also for the homogeneous tori case, we can similarly check that resonance
points of the discrete case (4.13) converge to that of the smooth case:

$\displaystyle\lim_{M,N\to\infty}\nu$
$\displaystyle=\lim_{M,N\to\infty}\frac{1}{4}\left(\cot^{2}\frac{\pi}{N}\tan^{2}\frac{k_{2}\pi}{\rho_{2}N}-\cot^{2}\frac{\pi}{M}\tan^{2}\frac{k_{1}\pi}{\rho_{1}M}\right)$
$\displaystyle=\frac{1}{4}\left(\frac{k_{2}^{2}}{\rho_{2}^{2}}-\frac{k_{1}^{2}}{\rho_{1}^{2}}\right).$

In fact, it follows that even the radii of the discrete homogeneous tori
admitting isothermic tori as Darboux transforms converge to the radii of the
smooth case. We can check this explicitly by noting that

$\lim_{M,N\to\infty}(1-4\nu
p^{2})=\frac{k_{1}^{2}}{\rho_{1}^{2}},\quad\lim_{M,N\to\infty}(1+4\nu
q^{2})=\frac{k_{2}^{2}}{\rho_{2}^{2}},$

which is consistent with the smooth case in [hertrich-
jeromin_introduction_2003, Equation (5.24)].

These observations suggest that the monodromy problem of Darboux transforms of
smooth isothermic surfaces can be investigated via structure preserving
discretisation without any numerical integration.

## References