100

# On the total CR twist of
transversal curves in the 3-sphere

Emilio Musso  (E. Musso) Dipartimento di Scienze Matematiche, Politecnico di
Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy
<EMAIL_ADDRESS>and  Lorenzo Nicolodi  (L. Nicolodi) Dipartimento di
Scienze Matematiche, Fisiche e Informatiche, Università di Parma, Parco Area
delle Scienze 53/A, I-43124 Parma, Italy<EMAIL_ADDRESS>Dedicated
to Peter Olver on the occasion of his 70th birthday

(Date: Version of June 20, 2023)

###### Abstract.

We investigate the total CR twist functional on transversal curves in the
standard CR 3-sphere $\mathrm{S}^{3}\subset\mathbb{C}^{2}$. The question of
the integration by quadratures of the critical curves and the problem of
existence and properties of closed critical curves are addressed. A procedure
for the explicit integration of general critical curves is provided and a
characterization of closed curves within a specific class of general critical
curves is given. Experimental evidence of the existence of infinite countably
many closed critical curves is provided.

###### 2010 Mathematics Subject Classification:

53C50; 53C42; 53A10

Authors partially supported by PRIN 2017 “Real and Complex Manifolds:
Topology, Geometry and holomorphic dynamics” (protocollo 2017JZ2SW5-004); and
by the GNSAGA of INdAM. The present research was also partially supported by
MIUR grant “Dipartimenti di Eccellenza” 2018-2022, CUP: E11G18000350001,
DISMA, Politecnico di Torino

##  1\. Introduction

The present paper finds its inspiration and theoretical framework in the
subjects of moving frames, differential invariants, and invariant variational
problems, three of the many research topics to which Peter Olver has made
lasting contributions. Among the many publications of Peter Olver dedicated to
these subjects, we like to mention [15, 16, 33, 34] as the ones that most
influenced our research activity.

More specifically, in this paper we further develop some of the themes
considered in [25, 30, 31] concerning the Cauchy-Riemann (CR) geometry of
transversal and Legendrian curves in the 3-sphere. In three dimensions, a CR
structure on a manifold is defined by an oriented contact distribution
equipped with a complex structure. While the automorphism group of a contact
manifold is infinite dimensional, that of a CR threefold is finite dimensional
and of dimension less or equal than eight [4, 5, 6]. The maximally symmetric
CR threefold is the 3-sphere ${\mathrm{S}}^{3}$, realized as a real
hyperquadric of $\mathbb{CP}^{2}$ acted upon transitively by the Lie group
$G\cong\mathrm{SU}(2,1)$. This homogeneous model allows the application of
differential-geometric techniques to the study of transversal and Legendrian
curves in ${\mathrm{S}}^{3}$. Since the seminal work of Bennequin [1], the
study of the topological properties of transversal and Legendrian knots in
3-dimensional contact manifolds has been an important area of research (see,
for instance, [9, 11, 13, 14, 12, 17] and the literature therein). Another
reason of interest for 3-dimensional contact geometry comes from its
applications to neuroscience. In fact, as shown by Hoffman [20], the visual
cortex can be modeled as a bundle equipped with a contact structure. For more
details, the interested reader is referred to the monograph [35, Chapter 5].
Recently, the CR geometry of Legendrian and transversal curves in
${\mathrm{S}}^{3}$ has also found interesting applications in the framework of
integrable system [3].

Let us begin by recalling some results from the CR geometry of transversal
curves in $\mathrm{S}^{3}$. According to [30], away from CR inflection points,
a curve transversal to the contact distribution of $\mathrm{S}^{3}$ can be
parametrized by a natural pseudoconformal parameter $s$ and in this
parametrization it is uniquely determined, up to CR automorphisms, by two
local CR invariants: the CR bending $\kappa$ and the CR twist $\tau$. This was
achieved by developing the method of moving frames and by constructing a
canonical frame field along generic111I.e., with no CR inflection points.
transversal curves. Moreover, for closed transversal curves, we defined three
discrete global invariants, namely, the wave number, the CR spin, and the CR
turning number. Next, we investigated the total strain functional, defined by
integrating the strain element $ds$. We proved that the corresponding critical
curves have constant bending and twist, and hence arise as orbits of
1-parameter groups of CR automorphisms. Finally, closed critical curves are
shown to be transversal positive torus knots with maximal Bennequin number.

In the present paper, we consider the CR invariant variational problem for
generic transversal curves in $\mathrm{S}^{3}$ defined by the total CR twist
functional,

$\mathcal{W}(\gamma)=\int_{\gamma}\tau ds.$

Our purpose is to address both the question of the explicit integration of
critical curves and the problem of existence and properties of closed critical
curves of $\mathcal{W}$.

We now give a brief outline of the content and results of this paper. In
Section 2, we shortly describe the standard CR structure of the 3-sphere
$\mathrm{S}^{3}$, viewed as a homogeneous space of the group $G$, and collect
some preliminary material. We then recall the basic facts about the CR
geometry of transversal curves in $\mathrm{S}^{3}$ as developed in [30] (see
the description above). Moreover, besides the already mentioned discrete
global invariants for a closed transversal curve, we introduce a fourth global
invariant, the trace of the curve with respect to a spacelike line.

In Section 3, we apply the method of moving frames and the Griffiths approach
to the calculus of variations [18, 19, 21] to compute the Euler–Lagrange
equations of the total CR twist functional. We construct the momentum space of
the corresponding variational problem and find a Lax pair formulation for the
Euler–Lagrange equations satisfied by the critical curves. This is the content
of Theorem A, the first main result of the paper, whose proof occupies the
whole Section 3. As a consequence of Theorem A, to each critical curve we
associate a momentum operator, which is a fixed element of the $G$-module
$\mathfrak{h}$ of traceless selfadjoint endomorphisms of $\mathbb{C}^{2,1}$.
From the conservation of the momentum along a critical curve, we derive two
conservation laws, involving two real parameters $c_{1}$ and $c_{2}$. The pair
$\mathbf{c}=(c_{1},c_{2})$ is referred to as the modulus of the critical
curve.

In Section 4, we introduce the phase type of the modulus of a critical curve.
We then define the phase curve of a given modulus and the associated notion of
signature of a critical curve with that given modulus. For a generic modulus
$\mathbf{c}$, the phase type of $\mathbf{c}$ refers to the properties of the
roots of the quintic polynomial in principal form given by

${\rm
P}_{\mathbf{c}}(x)=x^{5}+\frac{3}{2}c_{2}x^{2}+27c_{1}x-\frac{27}{2}c_{1}^{2}\,.$

The phase curve of the modulus $\mathbf{c}$ is the real algebraic curve
defined by the equation $y^{2}={\rm P}_{\mathbf{c}}(x)$. The signature of a
critical curve $\gamma$ with modulus $\mathbf{c}$ and nonconstant twist
provides a parametrization of the connected components of the phase curve of
$\mathbf{c}$ by the twist of $\gamma$. Importantly, the periodicity of the
twist of $\gamma$ amounts to the compactness of the image of the signature of
$\gamma$. This will play a role in Sections 5 and 6, where the closedness
question for critical curves is addressed. Using the Klein formulae for the
icosahedral solutions of the quintic [23, 32, 37], the roots of ${\rm
P}_{\mathbf{c}}$ can be evaluated in terms of hypergeometric functions. As a
byproduct, we show that the twist and the bending of a critical curve can be
obtained by inverting incomplete hyperelliptic integrals of the first kind. We
further specialize our analysis by introducing the orbit type of the modulus
$\mathbf{c}$ of a critical curve $\gamma$. The orbit type of $\mathbf{c}$
refers to the spectral properties of the momentum associated to $\gamma$.
Depending on the phase type, the number of connected components of the phase
curves, and the orbit type, the critical curves are then divided into twelve
classes. The critical curves of only three of these classes have periodic
twist.

In Section 5, we show that a general critical curve (cf. Definition 5.1) can
be integrated by quadratures using the momentum of the curve. This is the
content of Theorem B, the second main result of the paper. Theorem B is then
specialized to one of the twelve classes of critical curves, the class
characterized by the compactness of the connected component of the phase curve
and by the existence of three distinct real eigenvalues of the momentum.
Theorem C, the third main result, shows that the critical curves of this
specific class can be explicitly written by inverting hyperelliptic integrals
of the first and third kind. We then examine the closure conditions and prove
that a critical curve in this class is closed if and only if certain complete
hyperelliptic integrals depending on the modulus of the curve are rational.
Finally, the relations between these rational numbers and the global CR
invariants mentioned above are discussed.

In the last section, Section 6, we develop convincing heuristic and numerical
arguments to support the claim that there exist infinite countably many
distinct congruence classes of closed critical curves. These curves are
uniquely determined by the four discrete geometric invariants: the wave
number, the CR spin, the CR turning number, and the trace with respect to the
spacelike $\lambda_{1}$-eigenspace of the momentum. Using numerical tools, we
construct and illustrate explicit examples of approximately closed critical
curves.

##  2\. Preliminaries

###  2.1. The standard CR structure on the 3-sphere

Let $\mathbb{C}^{2,1}$ denote $\mathbb{C}^{3}$ with the indefinite Hermitian
scalar product of signature $(2,1)$ given by

(2.1)
$\langle\mathbf{z},\mathbf{w}\rangle={{}^{t}\\!\overline{\mathbf{z}}}\,\mathbf{h}\,\mathbf{w},\quad{\mathbf{h}}=(h_{ij})=\begin{pmatrix}0&0&i\\\
0&1&0\\\ -i&0&0\end{pmatrix}.$

Following common terminology in pseudo-Riemannian geometry, a nonzero vector
${\bf z}\in\mathbb{C}^{2,1}$ is _spacelike_ , _timelike_ or _lightlike_ ,
depending on whether $\langle{\bf z},{\bf z}\rangle$ is positive, negative or
zero. By $\mathcal{N}$ we denote the nullcone, i.e., the set of all lightlike
vectors.

Let $\mathcal{S}=\mathbb{P}(\mathcal{N})$ be the real hypersurface in
$\mathbb{C}P^{2}$ defined by

(2.2) $\mathcal{S}=\left\\{[{\bf
z}]\in\mathbb{C}P^{2}\mid\langle\mathbf{z},\mathbf{z}\rangle=i(\overline{z}_{1}z_{3}-\overline{z}_{3}z_{1})+\overline{z}_{2}z_{2}=0\right\\}.$

The restriction of the affine chart

(2.3)
$s:\mathbb{C}^{2}\ni(z_{1},z_{2})\longmapsto\left[{{}^{t}\\!\Big{(}}\frac{1+z_{1}}{2},i\frac{z_{2}}{\sqrt{2}},i\frac{1-z_{1}}{2}\Big{)}\right]\in\mathcal{S}\subset\mathbb{C}P^{2}$

to the unit sphere ${\mathrm{S}}^{3}$ of $\mathbb{C}^{2}$ defines a smooth
diffeomorphism between ${\mathrm{S}}^{3}$ and $\mathcal{S}$. For each $p=[{\bf
z}]\in\mathcal{S}$, the differential $(1,0)$-form

(2.4) $\tilde{\zeta}\big{|}_{p}=-\frac{i\langle\bf{z},d{\bf
z}\rangle}{\overline{{\bf z}}\,{{}^{t}\\!\bf
z}}\Big{|}_{p}\in\Omega^{1,0}(\mathbb{C}P^{2})\big{|}_{p}$

is well defined. In addition, the null space of the imaginary part of
$\tilde{\zeta}|_{p}$ is ${{T}(\mathcal{S})}|_{p}$, namely the tangent space of
$\mathcal{S}$ at $p$. Thus, the restriction of $\tilde{\zeta}$ to
${{T}(\mathcal{S})}$ is a real-valued 1-form
$\zeta\in\Omega^{1}(\mathcal{S})$. Since the pullback of $\zeta$ by the
diffeomorphism $s:\mathrm{S}^{3}\to\mathcal{S}$ is the standard contact form
$i\overline{{\bf z}}\cdot d{{\bf z}}|_{\mathrm{S}^{3}}$ of $\mathrm{S}^{3}$,
then $\zeta$ is a contact form whose contact distribution $\mathcal{D}$ is, by
construction, a complex subbundle of ${{T}(\mathbb{C}P^{2})|_{\mathcal{S}}}$.
Therefore, $\mathcal{D}$ inherits from
${{T}(\mathbb{C}P^{2})\big{|}_{\mathcal{S}}}$ a complex structure $J$. This
defines a CR structure on $\mathcal{S}$.

Let $\mathbf{e}_{1}$, $\mathbf{e}_{2}$, $\mathbf{e}_{3}$ denote the standard
basis of $\mathbb{C}^{3}$. Consider ${P}_{0}=[\mathbf{e}_{1}]\in{\mathcal{S}}$
and ${P}_{\infty}=[\mathbf{e}_{3}]\in{\mathcal{S}}$ as the origin and the
point at infinity of $\mathcal{S}$. Then,
$\dot{\mathcal{S}}:=\mathcal{S}\setminus\\{P_{\infty}\\}$ can be identified
with Euclidean 3-space with its standard contact structure $dz-ydx+xdy$ by
means of the Heisenberg projection222This map is the analogue of the
stereographic projection in Möbius (conformal) geometry.

$\pi_{H}:\dot{\mathcal{S}}\ni[{\bf
z}]\longmapsto{{}^{t}\\!\left(\mathrm{Re}(z_{2}/z_{1}),\mathrm{Im}(z_{2}/z_{1}),\mathrm{Re}(z_{3}/z_{1})\right)}\in\mathbb{R}^{3}.$

The inverse of the Heisenberg projection is the Heisenberg chart

$j_{H}:\mathbb{R}^{3}\ni{{}^{t}\\!(}x,y,z)\longmapsto\left[{{}^{t}\\!\big{(}}1,x+iy,z+\frac{i}{2}(x^{2}+y^{2})\big{)}\right]\in\dot{\mathcal{S}}.$

The Heisenberg chart can be lifted to a map whose image is a 3-dimensional
closed subgroup $\mathbb{H}^{3}$ of ${G}$, which is isomorphic to the
3-dimensional Heisenberg group [30].

Let $G$ be the special pseudo-unitary group of (2.1), i.e., the 8-dimensional
Lie group of unimodular complex $3\times 3$ matrices preserving (2.1),

(2.5)
$G=\\{A\in\mathrm{SL}(3,\mathbb{C})\mid{{}^{t}\\!\bar{A}}\mathbf{h}A=\mathbf{h}\\}\cong\mathrm{SU}(2,1),$

and let $\mathfrak{g}$ denote the Lie algebra of $G$,

(2.6)
$\mathfrak{g}=\\{X\in\mathfrak{sl}(3,\mathbb{C})\mid{{}^{t}\\!\bar{X}}\mathbf{h}+\mathbf{h}X=0\\}.$

The Maurer–Cartan form of the group $G$ takes the form

(2.7)
$\vartheta=A^{-1}dA=\begin{pmatrix}\alpha_{1}^{1}+i\beta_{1}^{1}&-i\alpha_{3}^{2}-\beta_{3}^{2}&\alpha_{3}^{1}\\\
\alpha_{1}^{2}+i\beta_{1}^{2}&-2i\beta_{1}^{1}&\alpha_{3}^{2}+i\beta_{3}^{2}\\\
\alpha_{1}^{3}&i\alpha_{1}^{2}+\beta_{1}^{2}&-\alpha_{1}^{1}+i\beta_{1}^{1}\\\
\end{pmatrix},$

where the 1-forms
$(\alpha_{1}^{1},\beta_{1}^{1},\alpha_{1}^{2},\beta_{1}^{2},\alpha_{1}^{3},\alpha_{3}^{2},\beta_{3}^{2},\alpha_{3}^{1})$
form a basis of the dual Lie algebra ${\mathfrak{g}}^{*}$. The center of $G$
is $Z=\\{\varpi
I_{3}\mid\varpi\in\mathbb{C},\,\varpi^{3}=1\\}\cong\mathbb{Z}_{3}$, where
$I_{3}$ denotes the $3\times 3$ identity matrix. Let $[G]$ denote the quotient
Lie group $G/Z$ and for $A\in G$ let $[A]$ denote its equivalence class in
$[G]$. Thus $[A]=[B]$ if and only if $B=\varpi A$, for some cube root of unity
$\varpi$. For any $A\in G$, the column vectors $(A_{1},A_{2},A_{3})$ of $A$
form a basis of $\mathbb{C}^{2,1}$ satisfying $\langle
A_{i},A_{j}\rangle=h_{ij}$ and $\mathrm{det}(A_{1},A_{2},A_{3})=1$. Such a
basis is referred to as a lightcone basis. On the other hand, a basis
$(\mathbf{u}_{1},\mathbf{u}_{2},\mathbf{u}_{3})$ of $\mathbb{C}^{2,1}$, such
that $\mathrm{det}(\mathbf{u}_{1},\mathbf{u}_{2},\mathbf{u}_{3})=1$ and
$\langle\mathbf{u}_{i},\mathbf{u}_{j}\rangle=\delta_{ij}\epsilon_{j}$, where
$\epsilon_{1}=-1$, $\epsilon_{2}=\epsilon_{3}=1$, is referred to as a
unimodular pseudo-unitary basis.

The group $G$ acts transitively and almost effectively on the left of
$\mathcal{S}$ by

(2.8) $A[{\bf z}]=[A{\bf z}],\quad\forall\,\,A\in{G},\,\,\forall\,\,[{\bf
z}]\in\mathcal{S}.$

This action descends to an effective action of $[{G}]={G}/{Z}$ on
$\mathcal{S}$. It is a classical result of E. Cartan [4, 5, 6] that $[{G}]$ is
the group of CR automorphisms of $\mathcal{S}$.

If we choose $[\mathbf{e}_{1}]=[{{}^{t}\\!(}1,0,0)]\in\mathcal{S}$ as an
origin of $\mathcal{S}$, the natural projection

(2.9) $\pi_{\mathcal{S}}:{G}\ni A\mapsto
A[\mathbf{e}_{1}]=[A_{1}]\in\mathcal{S}$

makes ${G}$ into a (trivial) principal fiber bundle with structure group

(2.10) ${G}_{0}=\left\\{A\in{G}\mid
A[\mathbf{e}_{1}]=[\mathbf{e}_{1}]\right\\}.$

The elements of ${G}_{0}$ consist of all $3\times 3$ unimodular matrices of
the form

(2.11) ${X}(\rho,\theta,v,r)=\begin{pmatrix}\rho e^{i\theta}&-i\rho
e^{-i\theta}\bar{v}&e^{i\theta}(r-\frac{i}{2}\rho|v|^{2})\\\
0&e^{-2i\theta}&v\\\ 0&0&\rho^{-1}e^{i\theta},\\\ \end{pmatrix},$

where $v\in\mathbb{C}$, $r\in\mathbb{R}$, $0\leq\theta<2\pi$, and $\rho>0$.

######  Remark 2.1.

The left-invariant 1-forms $\alpha_{1}^{2}$, $\beta_{1}^{2}$, $\alpha_{1}^{3}$
are linearly independent and generate the semi-basic 1-forms for the
projection $\pi_{\mathcal{S}}:G\to\mathcal{S}$. So, if
$s:U\subseteq\mathcal{S}\to{G}$ is a local cross section of
$\pi_{\mathcal{S}}$, then
$\left(s^{*}\alpha_{1}^{3},s^{*}\alpha_{1}^{2},s^{*}\beta_{1}^{2}\right)$
defines a coframe on $U$ and $s^{*}\alpha_{1}^{3}$ is a positive contact form.

###  2.2. Transversal curves

######  Definition 2.2.

Let $\gamma:J\to\mathcal{S}$ be a smooth immersed curve. We say that $\gamma$
is _transversal_ (to the contact distribution $\mathcal{D})$ if the tangent
vector ${\gamma^{\prime}}(t)\not\in\mathcal{D}|_{\gamma(t)}$, for every $t\in
J$. The parametrization $\gamma$ is said to be _positive_ if
$\zeta({\gamma^{\prime}}(t))>0$, for every $t$ and for every positive contact
form compatible with the CR structure. From now on, we assume that the
parametrization of a transversal curve is positive.

######  Definition 2.3.

Let $\gamma:J\to\mathcal{S}$ be a smooth curve. A _lift_ of $\gamma$ is a map
$\Gamma:J\to\mathcal{N}$ into the nullcone
$\mathcal{N}\subset\mathbb{C}^{2,1}$, such that $\gamma(t)=[\Gamma(t)]$, for
every $t\in J$.

If $\Gamma$ is a lift, any other lift is given by $r\Gamma$, where $r$ is a
smooth complex-valued function, such that $r(t)\neq 0$, for every $t\in J$.
From the definition of the contact distribution, we have the following.

######  Proposition 2.4.

A parametrized curve $\gamma:J\to\mathcal{S}$ is transversal and positively
oriented if and only if $-i\langle\Gamma,\Gamma^{\prime}\rangle|_{t}>0$, for
every $t\in J$ and for every lift $\Gamma$.

######  Definition 2.5.

A _frame field along $\gamma:J\to\mathcal{S}$_ is a smooth map $A:J\to{G}$
such that $\pi_{\mathcal{S}}\circ A=\gamma$. Since the fibration
$\pi_{\mathcal{S}}$ is trivial, there exist frame fields along every
transversal curve. If $A=(A_{1},A_{2},A_{3})$ is a frame field along $\gamma$,
$A_{1}$ is a lift of $\gamma$.

Let $A$ be a frame field along $\gamma$. Then

$A^{-1}A^{\prime}=\begin{pmatrix}a_{1}^{1}+ib_{1}^{1}&-ia_{3}^{2}-b_{3}^{2}&a_{3}^{1}\\\
a_{1}^{2}+ib_{1}^{2}&-2ib_{1}^{1}&a_{3}^{2}+ib_{3}^{2}\\\
a_{1}^{3}&ia_{1}^{2}+b_{1}^{2}&-a_{1}^{1}+ib_{1}^{1}\end{pmatrix},$

where $a_{1}^{3}$ is a strictly positive real-valued function. Any other frame
field along $\gamma$ is given by $\tilde{A}=AX(\rho,\theta,v,r)$, where $\rho$
($\rho>0$), $\theta$, $r:J\to\mathbb{R}$, $v=p+iq:J\to\mathbb{C}$ are smooth
functions and $X(\rho,\theta,v,r):J\to{G}_{0}$ is as in (2.11). If we let

$\tilde{A}^{-1}\tilde{A}^{\prime}=\begin{pmatrix}\tilde{a}_{1}^{1}+i\tilde{b}_{1}^{1}&-i\tilde{a}_{3}^{2}-\tilde{b}_{3}^{2}&\tilde{a}_{3}^{1}\\\
\tilde{a}_{1}^{2}+i\tilde{b}_{1}^{2}&-2i\tilde{b}_{1}^{1}&\tilde{a}_{3}^{2}+i\tilde{b}_{3}^{2}\\\
\tilde{a}_{1}^{3}&i\tilde{a}_{1}^{2}+\tilde{b}_{1}^{2}&-\tilde{a}_{1}^{1}+i\tilde{b}_{1}^{1}\end{pmatrix},$

then

(2.12)
$\tilde{A}^{-1}\tilde{A}^{\prime}=X^{-1}A^{-1}A^{\prime}X+X^{-1}X^{\prime},$

which implies

$\tilde{a}_{1}^{3}=\rho^{2}a_{1}^{3},\quad\tilde{a}_{1}^{2}+i\tilde{b}_{1}^{2}=\rho
e^{3i\theta}(a_{1}^{2}+ib_{1}^{2})-\rho^{2}e^{2i\theta}(p+iq)a_{1}^{3}.$

From this it follows that along any parametrized transversal curve there
exists a frame field $A$ for which $a_{1}^{2}+ib_{1}^{2}=0$. Such a frame
field is said to be of first order.

######  Definition 2.6.

Let $\Gamma$ be a lift of a transversal curve $\gamma:J\to\mathcal{S}$. If

$\det(\Gamma,\Gamma^{\prime},\Gamma^{\prime\prime})\big{|}_{t_{0}}=0,$

for some $t_{0}\in J$, then $\gamma({t_{0}})$ is called a _CR inflection
point_. The notion of CR inflection point is independent of the lift $\Gamma$.
A transversal curve with no CR inflection points is said to be _generic_. The
notion of a CR inflection point is invariant under reparametrizations and
under the action of the group of CR automorphisms.

######  Remark 2.7.

If $A:J\to G$ is a frame field along a transversal curve $\gamma$, then
$\gamma({t_{0}})$ is a CR inflection point if
$\det(A_{1},A_{1}^{\prime},A_{1}^{\prime\prime})\big{|}_{t_{0}}=0$. A
transversal curve all of whose points are CR inflection points is called a
chain. The notion of chain on a CR manifold goes back to Cartan [4, 5] (see
also [22] and the literature therein).

If $\gamma$ is transversal and $\Gamma$ is one of its lifts, then the complex
plane $[\Gamma\wedge\Gamma^{\prime}]_{t}$ is of type $(1,1)$ and the set of
null complex lines contained in $[\Gamma\wedge\Gamma^{\prime}]_{t}$ is a chain
which is independent of the choice of the lift $\Gamma$. This chain, denoted
by $\mathcal{C}_{\gamma}|_{t}$, is called the _osculating chain_ of $\gamma$
at $\gamma(t)$. By construction, $\mathcal{C}_{\gamma}|_{t}$ is the unique
chain passing through $\gamma(t)$ and tangent to $\gamma$ at the contact point
$\gamma(t)$. For more details on the CR-geometry of transversal curves in the
3-sphere, we refer to [30]. As a basic reference for transversal knots and
their topological invariants in the framework of 3-dimensional contact
geometry, we refer to [13] and the literature therein.

###  2.3. The canonical frame and the local CR invariants

In the following, we will consider generic transversal curves.

######  Definition 2.8.

Let $\gamma$ be a generic transversal curve. A lift $\Gamma$ of $\gamma$, such
that

$\det(\Gamma,\Gamma^{\prime},\Gamma^{\prime\prime})=-1,$

is said to be a _Wilczynski lift_ (W-lift) of $\gamma$. If $\Gamma$ is a
Wilczynski lift, any other is given by $\varpi\Gamma$, where
$\varpi\in\mathbb{C}$ is a cube root of unity. The function

$a_{\gamma}=i\langle\Gamma,\Gamma^{\prime}\rangle^{-1}$

is smooth, real-valued, and independent of the choice of $\Gamma$. We call
$a_{\gamma}$ the _strain density_ of the parametrized transversal curve
$\gamma$. The linear differential form $ds=a_{\gamma}dt$ is called the
_infinitesimal strain_.

######  Proposition 2.9 ([30]).

The strain density and the infinitesimal strain are invariant under the action
of the CR transformation group. In addition, if $h:I\to J$ is a change of
parameter, then the infinitesimal strains $ds$ and $d\tilde{s}$ of $\gamma$
and $\tilde{\gamma}=\gamma\circ h$, respectively, are related by
$d\tilde{s}=h^{*}(ds)$.

###### Proof.

This proof corrects a few misprints contained in the original one. If
$A\in{G}$ and if $\Gamma$ is a Wilczynski lift of $\gamma$, then
$\hat{\Gamma}=A\Gamma$ is a Wilczynski lift of $\hat{\gamma}=A\gamma$. This
implies that $a_{\gamma}=a_{\hat{\gamma}}$. Next, consider a reparametrization
$\tilde{\gamma}=\gamma\circ h$ of $\gamma$. Then, $\Gamma^{*}=\Gamma\circ h$
is a lift of $\tilde{\gamma}$, such that

$\det\left(\Gamma^{*},(\Gamma^{*})^{\prime},(\Gamma^{*})^{\prime\prime}\right)=-(h^{\prime})^{3}.$

This implies that $\tilde{\Gamma}=(h^{\prime})^{-1}\Gamma^{*}$ is a Wilczynski
lift of $\tilde{\gamma}$. Hence

$\langle\tilde{\Gamma},(\tilde{\Gamma})^{\prime}\rangle=(h^{\prime})^{-1}\langle\Gamma,\Gamma^{\prime}\rangle\circ
h.$

Therefore, the strain densities of $\gamma$ and $\tilde{\gamma}$ are related
by $a_{\tilde{\gamma}}=h^{\prime}a_{\gamma}\circ h$. Consequently, we have
$h^{*}(ds)=h^{\prime}a_{\gamma}\circ h\ dt=a_{\tilde{\gamma}}dt=d\tilde{s}$. ∎

As a straightforward consequence of Proposition 2.9, we have the following.

######  Corollary 2.10.

A generic transversal curve $\gamma$ can be parametrized so that
$a_{\gamma}=1$.

######  Definition 2.11.

If $a_{\gamma}=1$, we say that $\gamma:J\to\mathcal{S}$ is a _natural
parametrization_ , or a parametrization by the pseudoconformal strain or
pseudoconformal parameter. In the following, the natural parameter will be
denoted by $s$.

We can state the following.

######  Proposition 2.12 ([30]).

Let $\gamma:J\to\mathcal{S}$ be a generic transversal curve, parametrized by
the natural parameter. There exists a (first order) frame field $\mathcal{F}$
$=$ $(F_{1}$, $F_{2}$, $F_{3}):J\to{G}$ along $\gamma$, such that $F_{1}$ is a
W-lift and

(2.13) $\mathcal{F}^{-1}\mathcal{F}^{\prime}=\begin{pmatrix}i\kappa&-i&\tau\\\
0&-2i\kappa&1\\\ 1&0&i\kappa\\\ \end{pmatrix}=:K_{\kappa,\tau}(s),$

where $\kappa$, $\tau:J\to\mathbb{R}$ are smooth functions, called the _CR
bending_ and the _CR twist_ , respectively. The frame field $\mathcal{F}$ is
called a Wilczynski frame. If $\mathcal{F}$ is a Wilczynski frame, any other
is given by $\varpi\mathcal{F}$, where $\varpi$ is a cube root of unity. Thus,
there exists a unique frame field $[\mathcal{F}]:J\to[G]$ along $\gamma$,
called the canonical frame of $\gamma$.

######  Remark 2.13.

Given two smooth functions $\kappa$, $\tau:J\to\mathbb{R}$, there exists a
generic transversal curve $\gamma:J\to\mathcal{S}$, parametrized by the
natural parameter, whose bending is $\kappa$ and whose twist is $\tau$. The
curve $\gamma$ is unique up to CR automorphisms of $\mathcal{S}$.

######  Remark 2.14 (cf. [30]).

(1) Let $\gamma:J\to\mathcal{S}$ be as above and
$\mathcal{F}=(F_{1},F_{2},F_{3}):J\to{G}$ be a Wilczynski frame along
$\gamma$. Then,

$\gamma^{\\#}:J\ni s\mapsto[F_{3}(s)]_{\mathbb{C}}\in\mathcal{S}$

is an immersed curve, called the _dual_ of $\gamma$. The dual curve is
Legendrian (i.e., tangent to the contact distribution) if and only if
$\tau=0$. Thus, the twist can be viewed as a measure of how the dual curve
differs from being a Legendrian curve.

(2) Generic transversal curves with constant bending and twist have been
studied by the authors in [30]. In the following we will consider generic
transversal curves with nonconstant CR invariant functions.

###  2.4. Discrete CR invariants of a closed transversal curve

Referring to [30, 31], we briefly recall some CR invariants for closed
transversal curves, namely the notions of wave number, CR spin, and CR turning
number (or Maslov index). These invariants will be used in Sections 5 and 6.
The wave number is the ratio between the least period $\omega_{\gamma}$ of
$\gamma$ and the least period $\omega$ of the functions $(\kappa,\tau)$. The
CR spin is the ratio between $\omega_{\gamma}$ and the least period of a
Wilczynski lift of $\gamma$. The CR turning number is the degree (winding
number) of the map
$F_{1}-iF_{3}:\mathbb{R}/\omega_{\gamma}\mathbb{Z}\cong\mathrm{S}^{1}\to\dot{\mathbb{C}}\>=\mathbb{C}\setminus\\{0\\}$,
where $\mathcal{F}=(F_{1},F_{2},F_{3})$ is a Wilczynski frame along $\gamma$.

We will also make use of another invariant.

######  Definition 2.15.

Let $[{\bf z}]\in\mathbb{CP}^{2}$ be a spacelike line. Denote by
${\mathtt{C}}_{[{\bf z}]}$ the chain of all null lines orthogonal to $[{\bf
z}]$, equipped with its positive orientation. Consider a closed generic
transversal curve $\gamma$ with its positive orientation. Since $\gamma$ is
closed and generic, the intersection of $\gamma$ with ${\mathtt{C}}_{[{\bf
z}]}$ is either a finite set of points, or the empty set. The _trace of
$\gamma$ with respect to $[{\bf z}]$_, denoted by ${\rm
tr}_{[{\mathbf{z}}]}(\gamma)$, is the integer defined as follows: (1) if
$\gamma\cap{\mathtt{C}}_{[{\bf z}]}\neq\emptyset$, then ${\rm tr}_{[{\bf
z}]}(\gamma)$ counts the number of intersection points of $\gamma$ with
${\mathtt{C}}_{[{\bf z}]}$ (since $\gamma$ is not necessarily a simple curve,
the intersection points are counted with their multiplicities); (2) otherwise,
${\rm tr}_{[{\bf z}]}(\gamma)={\rm Lk}(\gamma,{\mathtt{C}}_{[{\bf z}]})$, the
linking number of $\gamma$ with ${\mathtt{C}}_{[\mathbf{z}]}$. The trace of
$\gamma$ is a $G$-equivariant map, that is, ${\rm tr}_{[{\bf z}]}(\gamma)={\rm
tr}_{{A}[{\bf z}]}({A}\gamma)$, for every ${A}\in G$.

##  3\. The total CR twist functional

Let $\mathfrak{T}$ be the space of generic transversal curves in
$\mathcal{S}$, parametrized by the natural parameter. We consider the total CR
twist functional $\mathcal{W}:\mathfrak{T}\to\mathbb{R}$, defined by

(3.1) $\mathcal{W}[\gamma]=\int_{J_{\gamma}}{\tau_{\gamma}\,\eta_{\gamma}}\,,$

where ${J_{\gamma}}$ is the domain of definition of the transversal curve
$\gamma$, $\tau_{\gamma}$ is its twist, and $\eta_{\gamma}=ds_{\gamma}$ is the
infinitesimal strain of $\gamma$ (cf. Section 2.3).

A curve $\gamma\in\mathfrak{T}$ is said to be a critical curve in
$\mathcal{S}$ if it is a critical point of $\mathcal{W}$ when one considers
compactly supported variations through generic transversal curves.

The main result of this section is the following.

######  Theorem A.

Let $\gamma:J\to\mathcal{S}$ be a generic transversal curve parametrized by
the natural parameter. Then, $\gamma$ is a critical curve if and only if

(3.2) $L^{\prime}(s)=[L(s),K_{\kappa,\tau}(s)]\,,$

where

(3.3) $L=\begin{pmatrix}0&i\tau^{\prime}+3\left(1-\tau\kappa\right)&2i\tau\\\
\tau&0&\tau^{\prime}+3i\left(1-\tau\kappa\right)\\\ 3i&-i\tau&0\\\
\end{pmatrix}$

and $K_{\kappa,\tau}$ is defined as in (2.13).

###### Proof.

The proof of Theorem A is organized in four steps and three lemmas.

Step 1. We show that generic transversal curves are in 1-1 correspondence with
the integral curves of a suitable Pfaffian differential system.

Let $\gamma:J\to\mathcal{S}$ be a generic transversal curve parametrized by
the natural parameter. According to Proposition 2.12, the canonical frame of
$\gamma$ defines a unique lift $[\mathcal{F}]:J\to[G]$. The map

$\mathfrak{f}:J\ni
s\longmapsto\big{(}[\mathcal{F}(s)],\kappa(s),\tau(s)\big{)}\in[G]\times\mathbb{R}^{2}$

is referred to as the extended frame of $\gamma$. The product space
$M:=[G]\times\mathbb{R}^{2}$ is called the _configuration space_. The
coordinates on $\mathbb{R}^{2}$ will be denoted by $(\kappa,\tau)$.

With some abuse of notation, we use $\alpha_{1}^{1}$, $\beta_{1}^{1}$,
$\alpha_{1}^{2}$, $\beta_{1}^{2}$, $\alpha_{1}^{3}$, $\alpha_{3}^{2}$,
$\beta_{3}^{2}$, $\alpha_{3}^{1}$ to denote the entries of the Maurer–Cartan
form of $[{G}]$ as well as their pull-backs on the configuration space $M$. By
Proposition 2.12, the extended frames of $\gamma$ are the integral curves of
the Pfaffian differential system $(\mathcal{A},\eta)$ on $M$ generated by the
linearly independent 1-forms

$\displaystyle\mu^{1}=\alpha_{1}^{2},\quad\mu^{2}=\beta_{1}^{2},\quad\mu^{3}=\alpha_{3}^{2}-{\alpha^{3}_{1}},\quad\mu^{4}=\beta_{3}^{2},$
$\displaystyle\mu^{5}=\alpha_{1}^{1},\quad\mu^{6}=\beta_{1}^{1}-\kappa\alpha_{1}^{3},\quad\mu^{7}=\alpha_{3}^{1}-\tau\alpha_{1}^{3},$

with the independence condition $\eta:=\alpha_{1}^{3}$.

If $\mathfrak{f}=\big{(}[\mathcal{F}],\kappa,\tau\big{)}:J\to M$ is an
integral curve of $(\mathcal{A},\eta)$, then $\gamma=[F_{1}]:J\to\mathcal{S}$
defines a generic transversal curve, such that $[\mathcal{F}]$ is its
canonical frame, $\kappa$ its bending and $\tau$ its twist. Accordingly, the
integral curves of $(\mathcal{A},\eta)$ are the extended frames of generic
transversal curves in $\mathcal{S}$.

Thus, generic transversal curves are in 1-1 correspondence with the integral
curves of the Pfaffian system $(\mathcal{A},\eta)$ on the configuration space
$M$.

If we put

$\pi^{1}=d\kappa,\quad\pi^{2}=d\tau,$

the 1-forms $(\eta,\mu^{1},\dots,\mu^{7},\pi^{1},\pi^{2})$ define an absolute
parallelism on $M$. Exterior differentiation and use of the Maurer–Cartan
equations of ${G}$ yield the following structure equations for the coframe
$(\eta,\mu^{1},\mathellipsis,\mu^{7},\pi^{1},\pi^{2})$:

(3.4) $\begin{cases}d\eta=2\mu^{1}\wedge\mu^{2}+2\mu^{5}\wedge\eta,\\\
d\pi^{1}=d\pi^{2}=0,\\\ \end{cases}$ (3.5)
$\begin{cases}d\mu^{1}=-\mu^{1}\wedge\mu^{5}+3\mu^{2}\wedge\mu^{6}+(3\kappa\mu^{2}-\mu^{3})\wedge\eta,\\\
d\mu^{2}=-3\mu^{1}\wedge\mu^{6}-\mu^{2}\wedge\mu^{3}-(3\kappa\mu^{1}+\mu^{4})\wedge\eta,\\\
d\mu^{3}=-2\mu^{1}\wedge\mu^{2}-\mu^{1}\wedge\mu^{7}+\mu^{3}\wedge\mu^{5}+3\mu^{4}\wedge\mu^{6}\\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-(\tau\mu^{1}-3\kappa\mu^{4}+3\mu^{5})\wedge\eta,\\\
d\mu^{4}=-\mu^{2}\wedge\mu^{7}-3\mu^{3}\wedge\mu^{6}+\mu^{4}\wedge\mu^{5}-(\tau\mu^{2}+3\kappa\mu^{3}-3\mu^{6})\wedge\eta,\\\
d\mu^{5}=-\mu^{1}\wedge\mu^{4}+\mu^{2}\wedge\mu^{3}+(\mu^{2}-\mu^{7})\wedge\eta,\\\
d\mu^{6}=-2\kappa\mu^{1}\wedge\mu^{2}-\mu^{1}\wedge\mu^{3}-\mu^{2}\wedge\mu^{4}-(\mu^{1}+2\kappa\mu^{5})\wedge\eta-\pi^{1}\wedge\eta,\\\
d\mu^{7}=-2\tau\mu^{1}\wedge\mu^{2}-2\mu^{3}\wedge\mu^{4}-2\mu^{5}\wedge\mu^{7}\\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+(2\mu^{4}-2\tau\mu^{5})\wedge\eta-\pi^{2}\wedge\eta.\\\
\end{cases}$

######  Remark 3.1.

From the structure equations it follows that the derived flag of
$(\mathcal{A},\eta)$ is given by
$\mathcal{A}_{(4)}\subset\mathcal{A}_{(3)}\subset\mathcal{A}_{(2)}\subset\mathcal{A}_{(1)}$,
where $\mathcal{A}_{(4)}=\\{0\\}$,
$\mathcal{A}_{(3)}=\text{span}\\{\mu^{1}\\}$,
$\mathcal{A}_{(2)}=\text{span}\\{\mu^{1},\mu^{2},\mu^{3}\\}$,
$\mathcal{A}_{(1)}$ $=$ $\text{span}\\{\mu^{1}$, $\mu^{2}$, $\mu^{3}$,
$\mu^{4}$, $\mu^{5}\\}$. Thus, all the derived systems of $(\mathcal{A},\eta)$
have constant rank. For the notion of derived flag, see [19].

Step 2. We develop a construction due to Griffiths [19] on an affine subbundle
of $T^{*}(M)$ (cf. also [2, 18, 21]) in order to derive the Euler–Lagrange
equations.

Let $\mathcal{Z}\subset T^{*}(M)$ be the affine subbundle defined by the
1-forms $\mu^{1},\dots,\mu^{7}$ and $\lambda:=\tau\eta$, namely

$\mathcal{Z}=\lambda+\text{span}\\{\mu^{1},\dots,\mu^{7}\\}\subset T^{*}(M).$

We call $\mathcal{Z}$ the phase space of the Pfaffian system
$(\mathcal{A},\eta)$. The 1-forms $(\mu^{1},\dots,\mu^{7},{\lambda})$ induce a
global affine trivialization of $\mathcal{Z}$, which may be identified with
$M\times\mathbb{R}^{7}$ by the map

$M\times\mathbb{R}^{7}\ni\left(([\mathcal{F}],\kappa,\tau),p_{1},\dots,p_{7}\right)\longmapsto{\lambda_{|_{([\mathcal{F}],\kappa,\tau)}}}+\sum_{j=1}^{7}p_{j}{\mu^{j}}_{|_{([\mathcal{F}],\kappa,\tau)}}\in\mathcal{Z}\,,$

where $p_{1},\dots,p_{7}$ are the fiber coordinates of the bundle map
$\mathcal{Z}\to M$ with respect to the trivialization. Under this
identification, the restriction to $\mathcal{Z}$ of the Liouville (canonical)
1-form of $T^{*}(M)$ takes the form

$\xi=\tau\,\eta+\sum_{j=1}^{7}p_{j}\mu^{j}\,.$

Exterior differentiation and use of the quadratic equations (3.4) and (3.5)
yield

$\begin{split}d\xi&\equiv\pi^{2}\wedge\eta+2\tau\mu^{5}\wedge\eta+\sum_{j=1}^{7}dp_{j}\wedge\mu^{j}+p_{1}(3\kappa\mu^{2}-\mu^{3})\wedge\eta\\\
&\quad-p_{2}(3\kappa\mu^{1}+\mu^{4})\wedge\eta-
p_{3}(\tau\mu^{1}-3\kappa\mu^{4}+3\mu^{5})\wedge\eta\\\ &\quad-
p_{4}(\tau\mu^{2}+3\kappa\mu^{3}-3\mu^{6})\wedge\eta+p_{5}(\mu^{2}-\mu^{7})\wedge\eta\\\
&\quad-p_{6}(\pi^{1}+\mu^{1}+2\kappa\mu^{5})\wedge\eta-
p_{7}(\pi^{2}-2\mu^{4}+4\tau\mu^{5})\wedge\eta\,,\end{split}$

where the sign ‘$\equiv$’ denotes equality modulo the span of
$\\{\mu^{i}\wedge\mu^{j}\\}_{i,j=1,\dots,7}$ . The Cartan system
$(\mathcal{C}(d\xi),\eta)$ of the 2-form $d\xi$ is the Pfaffian system on
$\mathcal{Z}$ generated by the 1-forms

$\left\\{X{\lrcorner}\,d\xi\mid
X\in\mathfrak{X}(\mathcal{Z})\right\\}\subset\Omega^{1}(\mathcal{Z}),$

with independence condition $\eta\neq 0$.

By Step 1, generic transversal curves are in 1-1 correspondence with the
integral curves of the Pfaffian system $(\mathcal{A},\eta)$.

Let $\mathfrak{f}:J\to M$ be the extended frame corresponding to the generic
transversal curve $\gamma:J\to\mathcal{S}$ parametrized by the natural
parameter. According to Griffiths approach to the calculus of variations (cf.
[2, 18, 19, 21]), if the extended frame $\mathfrak{f}$ admits a lift
$y:J\to\mathcal{Z}$ to the phase space $\mathcal{Z}$ which is an integral
curve of the Cartan system $(\mathcal{C}(d\xi),\eta)$, then $\gamma$ is a
critical curve of the total twist functional with respect to compactly
supported variations.

As observed by Bryant [2], if all the derived systems of $(\mathcal{A},\eta)$
are of constant rank, as in the case under discussion (cf. Remark 3.1), then
the converse is also true. Hence all extremal trajectories arise as
projections of integral curves of the Cartan system
$(\mathcal{C}(d\xi),\eta)$.

Next, we compute the Cartan system $(\mathcal{C}(d\xi),\eta)$. Contracting the
2-form $d\xi$ with the vector fields of the tangent frame

$(\partial_{\eta},\partial_{\mu^{1}},\dots,\partial_{\mu^{7}},\partial_{\pi^{1}},\partial_{\pi^{2}},\partial_{p_{1}},\mathellipsis,\partial_{p_{7}})$

on $\mathcal{Z}$, dual to the coframe
$(\eta,\mu^{1},\dots,\mu^{7},\pi^{1},\pi^{2},dp_{1},\dots,dp_{7})$, yields the
1-forms

(3.6) $\partial_{p_{j}}{\lrcorner}\,d\xi\equiv\mu^{j},\quad j=1,\dots,7,\\\ $

(3.7) $\begin{cases}-\partial_{\pi^{1}}\lrcorner\,d\xi\equiv
p_{6}\eta=:\dot{\pi}_{1},\\\
-\partial_{\pi^{2}}\lrcorner\,d\xi\equiv(p_{7}-1)\eta=:\dot{\pi}_{2},\\\
-\partial_{\eta}\lrcorner\,d\xi\equiv(1-p_{7})\pi^{2}=:\dot{\eta},\\\
\end{cases}$

(3.8) $\begin{cases}-\partial_{\mu^{1}}\lrcorner\,d\xi\equiv dp_{1}+(3\kappa
p_{2}+\tau p_{3}+p_{6})\eta=:\dot{\mu}^{1},\\\
-\partial_{\mu^{2}}\lrcorner\,d\xi\equiv dp_{2}-(3\kappa p_{1}-\tau
p_{4}+p_{5})\eta=:\dot{\mu}^{2},\\\ -\partial_{\mu^{3}}\lrcorner\,d\xi\equiv
dp_{3}+(p_{1}+3\kappa p_{4})\eta=:\dot{\mu}^{3},\\\
-\partial_{\mu^{4}}\lrcorner\,d\xi\equiv dp_{4}+(p_{2}-3\kappa
p_{3}-2p_{7})\eta=:\dot{\mu}^{4},\\\ -\partial_{\mu^{5}}\lrcorner\,d\xi\equiv
dp_{5}-(2\tau-3p_{3}-2\kappa p_{6}-4\tau p_{7})\eta=:\dot{\mu}^{5},\\\
-\partial_{\mu^{6}}\lrcorner\,d\xi\equiv dp_{6}-3p_{4}\eta=:\dot{\mu}^{6},\\\
-\partial_{\mu^{7}}\lrcorner\,d\xi\equiv dp_{7}+p_{5}\eta=:\dot{\mu}^{7}.\\\
\end{cases}$

We have proved the following.

######  Lemma A1.

The Cartan system $(\mathcal{C}(d\xi),\eta)$ is the Pfaffian system on
$\mathcal{Z}\cong M\times\mathbb{R}^{7}$ generated by the 1-forms

$\\{\mu^{1},\dots,\mu^{7},\dot{\pi}_{1},\dot{\pi}_{2},\dot{\eta},\dot{\mu}^{1},\dots,\dot{\mu}^{7}\\}$

and with independence condition $\eta\neq 0$.

Now, the Cartan system $(\mathcal{C}(d\xi),\eta)$ is reducible, i.e., there
exists a nonempty submanifold $\mathcal{Y}\subseteq\mathcal{Z}$, called the
reduced space, such that: (1) at each point of $\mathcal{Y}$ there exists an
integral element of $(\mathcal{C}(d\xi),\eta)$ tangent to $\mathcal{Y}$; (2)
if $\mathcal{X}\subseteq\mathcal{Z}$ is any other submanifold with the same
property of $\mathcal{Y}$, then $\mathcal{X}\subseteq\mathcal{Y}$. The reduced
space $\mathcal{Y}$ is called the momentum space of the variational problem.
Moreover, the restriction of the Cartan system $(\mathcal{C}(d\xi),\eta)$ to
$\mathcal{Y}$ is called the Euler–Lagrange system of the variational problem,
and will be denoted by $(\mathcal{J},\eta)$.

A basic result states that the Pfaffian systems $(\mathcal{C}(d\xi),\eta)$ and
$(\mathcal{J},\eta)$ have the same integral curves (cf. [18, 19]).

The system $(\mathcal{J},\eta)$ can be constructed by an algorithmic procedure
(cf. [19]).

######  Lemma A2.

The momentum space $\mathcal{Y}$ is the 11-dimensional submanifold of
$\mathcal{Z}$ defined by the equations

$p_{7}=1,\quad p_{6}=p_{5}=p_{4}=0,\quad p_{3}=-\frac{2}{3}\tau\,\quad
p_{2}=2(1-\tau\kappa)\,.$

The Euler–Lagrange system $(\mathcal{J},\eta)$ is the Pfaffian system on
$\mathcal{Y}\cong M\times\mathbb{R}$, with independence condition $\eta\neq
0$, generated by the 1 forms

(3.9)
$\begin{cases}{\mu^{1}}_{|\mathcal{Y}},\dots,{\mu^{7}}_{|\mathcal{Y}},\\\
\sigma_{1}=dp_{1}+6\kappa(1-\tau\kappa)\eta-\frac{2}{3}\tau^{2}\eta\,,\\\
\sigma_{2}=-2\tau d\kappa-2\kappa d\tau-3kp_{1}\eta\,,\\\
\sigma_{3}=-2d\tau+3p_{1}\eta\,.\end{cases}$

###### Proof of Lemma A2.

Let $V_{1}(d\xi)\hookrightarrow\mathbb{P}(T(\mathcal{Z}))\to\mathcal{Z}$ be
the totality of 1-dimensional integral elements of $(\mathcal{C}(d\xi),\eta)$.
In view of (3.7), we find that

${V_{1}(d\xi)}_{|_{(([\mathcal{F}],\kappa,\tau);p_{1},\dots,p_{7})}}\neq\emptyset\iff
p_{6}=0,\quad p_{7}=1.$

Thus, the image $\mathcal{Z}_{1}\subset\mathcal{Z}$ of $V_{1}(d\xi)$ with
respect to the natural projection $V_{1}(d\xi)\to\mathcal{Z}$, is given by

$\mathcal{Z}_{1}=\left\\{(([\mathcal{F}],\kappa,\tau);p_{1},\dots,p_{7})\in\mathcal{Z}\mid
p_{6}=0,\,p_{7}=1\right\\}.$

Next, the restriction of $\dot{\mu}^{6}$ and $\dot{\mu}^{7}$ to
$\mathcal{Z}_{1}$ take the form $\dot{\mu}^{6}=-3p_{4}\eta$ and
$\dot{\mu}^{7}=p_{5}\eta$. Thus, the second reduction $\mathcal{Z}_{2}$ is
given by

$\mathcal{Z}_{2}=\left\\{(([\mathcal{F}],\kappa,\tau);p_{1},\dots,p_{7})\in\mathcal{Z}_{1}\mid
p_{4}=p_{5}=0\right\\}.$

Considering the restriction of $\dot{\mu}^{4}$ and $\dot{\mu}^{5}$ to
$\mathcal{Z}_{2}$ yields the equations

$p_{2}=2(1-\tau\kappa),\quad p_{3}=-\frac{2}{3}\tau\,,$

which define the third reduction $\mathcal{Z}_{3}$. Now, the restriction
$\mathcal{C}_{3}(d\xi)$ to $\mathcal{Z}_{3}$ of the Cartan system
$\mathcal{C}(d\xi)$ is generated by the 1-forms $\mu^{1}$, $\dots$, $\mu^{7}$
and

$\begin{cases}\sigma_{1}=dp_{1}+6\kappa(1-\tau\kappa)\eta-\frac{2}{3}\tau^{2}\eta\,,\\\
\sigma_{2}=dp_{2}-3kp_{1}\eta=-2\tau d\kappa-2\kappa d\tau-3kp_{1}\eta\,,\\\
\sigma_{3}=-2d\tau+3p_{1}\eta\,.\end{cases}$

This implies that there exists an integral element of $V_{1}(d\xi)$ over each
point of $\mathcal{Z}_{3}$, i.e., ${V_{1}(d\xi)}_{|p}\neq\emptyset$, for each
$p\in\mathcal{Z}_{3}$. Hence, $\mathcal{Y}:=\mathcal{Z}_{3}$ is the momentum
space and $(\mathcal{J},\eta):=(\mathcal{C}_{3}(d\xi),\eta)$ is the reduced
system of $(\mathcal{C}(d\xi),\eta)$. ∎

Step 3. We derive the Euler–Lagrange equations.

By the previous discussion, all the extremal trajectories of $\mathcal{S}$
arise as projections of the integral curves of the Euler–Lagrange system. If
$y:J\to\mathcal{Y}$ is an integral curve of the Euler–Lagrange system
$(\mathcal{J},\eta)$ and $\mathtt{pr}:\mathcal{Y}\to\mathcal{S}$ is the
natural projection of $\mathcal{Y}$ onto $\mathcal{S}$, then
$\gamma=\mathtt{pr}\circ y:J\to\mathcal{S}$ is a critical curve of the total
twist functional with respect to compactly supported variations.

We can prove the following.

######  Lemma A3.

A curve $y:J\to\mathcal{Y}$ is an integral curve of the Euler–Lagrange system
$(\mathcal{J},\eta)$ if and only if the bending $\kappa$ and the twist $\tau$
of the transversal curve $\gamma=\mathtt{pr}\circ y:J\to\mathcal{S}$ satisfy
the equations

(3.10) $\displaystyle 2\kappa\tau^{\prime}+\tau\kappa^{\prime}=0\,,$ (3.11)
$\displaystyle\tau^{\prime\prime}+9\kappa(1-\tau\kappa)-\tau^{2}=0\,.$

###### Proof of Lemma A3.

If $y=(([\mathcal{F}],\kappa,\tau);p_{1}):J\to\mathcal{Y}$ is an integral
curve of the Euler–Lagrange system $(\mathcal{J},\eta)$, the projection
$\gamma=\mathtt{pr}\circ y$ is the smooth curve $\gamma(s)=[{F}_{1}(s)]$,
where ${F}_{1}$ is the first column of $\mathcal{F}$. The equations

$\mu^{1}=\cdots=\mu^{7}=0$

together with the independence condition $\eta\neq 0$ tell us that
$([\mathcal{F}],\kappa,\tau)$ is an integral curve of the Pfaffian system
$(\mathcal{A},\eta)$ on the configuration space $M$. Hence $\gamma$ is a
generic transversal curve with bending $\kappa$, twist $\tau$ and
$\mathcal{F}$ is a Wilczynski frame along $\gamma$. Next, for the smooth
function $\kappa$, $\tau:J\to\mathbb{R}$, let $\kappa^{\prime}$,
$\kappa^{\prime\prime}$ and $\tau^{\prime}$, $\tau^{\prime\prime}$, etc., be
defined by

$d\kappa=\kappa^{\prime}\eta,\quad
d\kappa^{\prime}=\kappa^{\prime\prime}\eta,\quad d\tau=\tau^{\prime}\eta,\quad
d\tau^{\prime}=\tau^{\prime\prime}\eta\,.$

With reference to (3.9), equation $\sigma_{3}=0$ implies

$p_{1}=\frac{2}{3}\tau^{\prime}\,.$

Further, $\sigma_{2}=0$ gives

$2\kappa\tau^{\prime}+\tau\kappa^{\prime}=0\,.$

Finally, equation $\sigma_{1}=0$ yields

$\tau^{\prime\prime}+9\kappa(1-\tau\kappa)-\tau^{2}=0\,.$

Conversely, let $\gamma:J\to\mathcal{S}$ be a generic transversal curve,
parametrized by the natural parameter, satisfying (3.10) and (3.11) and let
$[\mathcal{F}]$ its canonical frame. Then,

$y(s)=\Big{(}([\mathcal{F}],\kappa,\tau);\frac{2}{3}\tau^{\prime}\Big{)}$

is, by construction, an integral curve of the Euler–Lagrange system
$(\mathcal{J},\eta)$. ∎

Step 4. We eventually provide a Lax formulation for the Euler–Lagrange
equations (cf. (3.10) and (3.11)) of a critical curve
$\gamma:J\to\mathcal{S}$.

Using the Killing form of $\mathfrak{g}$, the dual Lie algebra
$\mathfrak{g}^{*}$ can be identified with $\mathfrak{h}=i\mathfrak{g}$, the
$G$-module of traceless selfadjoint endomorphisms of $\mathbb{C}^{2,1}$. Under
this identification, the restriction to $\mathcal{Y}$ of the tautological
1-form $\xi$ goes over to an element of $\mathfrak{h}$ which originates the
$\mathfrak{h}$-valued function $L:J\to\mathfrak{h}$ given by

(3.12)
$L(s)=\begin{pmatrix}0&i\tau^{\prime}+3\left(1-\tau\kappa\right)&2i\tau\\\
\tau&0&\tau^{\prime}+3i\left(1-\tau\kappa\right)\\\ 3i&-i\tau&0\\\
\end{pmatrix}.$

A direct computation shows that the Euler–Lagrange equations (3.10) and (3.11)
of the critical curve $\gamma$ are satisfied if and only if

$L^{\prime}(s)=\left[L(s),K_{\kappa,\tau}(s)\right]\,,$

where $K_{\kappa,\tau}$ is given by (2.13). This concludes the proof of
Theorem A. ∎

As a consequence of Theorem A, we have the following.

######  Corollary 3.2.

Let $\gamma:J\to\mathcal{S}$ be a generic transversal curve parametrized by
the natural parameter. Let $[\mathcal{F}]:J\to[G]$ be the canonical frame of
$\gamma$ and let $L:J\to\mathfrak{h}$ be as in (3.12). If $\gamma$ is a
critical curve, the Lax equation (3.2) implies that

(3.13)
$\mathcal{F}(s)\,L(s)\,\mathcal{F}^{-1}(s)=\mathfrak{M}\,,\quad\forall\,s\in
J.$

where $\mathfrak{M}$ is a fixed element of $\mathfrak{h}$.

######  Definition 3.3.

The element $\mathfrak{M}\in\mathfrak{h}$ is called the momentum of the
critical curve $\gamma$.

The characteristic polynomial of the momentum $\mathfrak{M}$ is

(3.14)
$-x^{3}-6\kappa\tau^{2}x+54\kappa\tau-27\kappa^{2}\tau^{2}+2\tau^{3}-3\tau^{\prime
2}-27\,.$

The conservation of the momentum along $\gamma$ yields the two conservation
laws

(3.15) $\displaystyle\kappa\tau^{2}={c_{1}}\,,$ (3.16)
$\displaystyle-18\kappa\tau+9\kappa^{2}\tau^{2}-\frac{2}{3}\tau^{3}+\tau^{\prime
2}=C_{2}-9\,,$

for real constants $c_{1}$ and $C_{2}$. We let $c_{2}:=C_{2}-9$. Using this
notation, the (opposite of the) characteristic polynomial of the momentum is

(3.17) ${\rm Q}(x)=x^{3}+6c_{1}x+(27+3c_{2}).$

If $c_{1}\neq 0$, the twist and the bending are never zero and the
conservation laws can be rewritten as

(3.18) $\kappa=c_{1}\tau^{-2},\quad\frac{3}{2}\tau^{2}\tau^{\prime
2}=\tau^{5}+\frac{3}{2}c_{2}\tau^{2}+27c_{1}\tau-\frac{27}{2}c_{1}^{2}.$

If $c_{1}=0$, it can be easily proved that $\kappa=0$ and the second
conservation law takes the form

(3.19) $\tau^{\prime 2}=\frac{2}{3}\tau^{3}+c_{2}.$

######  Definition 3.4.

The pair of real constants ${\bf c}=(c_{1},c_{2})$ is called the modulus of
the critical curve $\gamma$.

######  Remark 3.5.

For the application of Griffiths’ approach to other geometric variational
problems, the reader is referred to [8, 10, 18, 19, 26, 27, 29].

##  4\. The CR twist of a critical curve

###  4.1. Phase types

For ${\bf c}=(c_{1},c_{2})\in\mathbb{R}^{2}$, we denote by ${\rm P}_{{\bf c}}$
the quintic polynomial in principal form given by

(4.1) ${\rm P}_{{\bf
c}}(x)=x^{5}+\frac{3}{2}c_{2}x^{2}+27c_{1}x-\frac{27}{2}c_{1}^{2}$

and by ${\rm Q}_{{\bf c}}$ the cubic polynomial given by

(4.2) ${\rm Q}_{{\bf c}}(x)=x^{3}+6c_{1}x+(27+3c_{2}).$

Excluding the case ${\bf c}=0$, ${\rm P}_{{\bf c}}$ possesses at least a pair
of complex conjugate roots.

######  Definition 4.1.

We adopt the following terminology.

  * •

${\bf c}\in\mathbb{R}^{2}$ is of _phase type ${\mathcal{A}}$_ if ${\rm
P}_{{\bf c}}$ has four complex roots $a_{j}\pm ib_{j}$, $j=1,2$,
$0<b_{1}<b_{2}$, and a simple real root $e_{1}$;

  * •

${\bf c}\in\mathbb{R}^{2}$ is of _phase type ${\mathcal{B}}$_ if ${\rm
P}_{{\bf c}}$ has two complex roots $a\pm ib$, $b>0$, and three simple real
roots $e_{1}<e_{2}<e_{3}$;

  * •

${\bf c}\in\mathbb{R}^{2}$ is of _phase type_ ${\mathcal{C}}$ if ${\rm
P}_{{\bf c}}$ has a multiple real root.

In the latter case, two possibilities may occur: (1) ${\rm P}_{{\bf c}}$ has a
double real root and a simple real root; or (2) ${\rm P}_{{\bf c}}$ has a real
root of multiplicity $5$.

By the same letters, we also denote the corresponding sets of moduli of phase
types ${\mathcal{A}}$, ${\mathcal{B}}$, and ${\mathcal{C}}$, respectively.

Next, we give a more detailed description of the sets ${\mathcal{A}}$,
${\mathcal{B}}$, and ${\mathcal{C}}$. To this end, we start by defining the
_separatrix curve_. Let $(m,n)$ be the homogeneous coordinates of
$\mathbb{RP}^{1}$ and let $[(m_{*},n_{*})]$ be the point of $\mathbb{RP}^{1}$
such that $3m_{*}^{3}+6m_{*}^{2}n_{*}+4m_{*}n_{*}^{2}+2n_{*}^{3}=0$ (i.e.,
$m_{*}=1$ and $n_{*}\approx-0.72212$).

######  Definition 4.2.

The _separatrix curve_ $\Xi\subset\mathbb{R}^{2}$ is the image of the
parametrized curve
$\xi=(\xi_{1},\xi_{2}):\mathbb{RP}^{1}\setminus\\{[(m_{*},n_{*})]\\}\to\mathbb{R}^{2}$,
defined by

$\begin{split}{\xi_{1}([(m,n)])}&=\frac{6\sqrt[3]{2}mn^{4/3}(3m^{2}+2mn+n^{2})^{4/3}}{(3m^{3}+6m^{2}n+4mn^{2}+2n^{3})^{5/3}},\\\
{\xi_{2}([(m,n)])}&=-\frac{36n(3m^{2}+2mn+n^{2})(4m^{3}+3m^{2}n+2mn^{2}+n^{3})}{(3m^{3}+6m^{2}n+4mn^{2}+2n^{3})^{2}}.\end{split}$

######  Remark 4.3.

The map $\xi$ is injective and $\Xi$ has a cusp at
$\xi([(1,1)])=(\frac{4}{5}(\frac{6}{5})^{2/3},-\frac{48}{5}))$. It is regular
elsewhere. In addition, $\Xi$ has a horizontal inflection point at
${\xi([(0,1)])}=(0,-9)$. Let ${\rm J}_{\xi}$ be the interval
$(\arctan(n_{*}),\arctan(n_{*})+\pi)\approx(-0.625418,2.51617)$. Then,
$\widetilde{\xi}:t\in{\rm J}_{\xi}\to\xi(\cos(t),\sin(t))\in\Xi$ is another
parametrization of $\Xi$. The inflection point is $\widetilde{\xi}(\pi/2)$.
The “negative part” $\Xi_{-}=\Xi\cap\\{{\bf c}\in\mathbb{R}^{2}\mid
c_{1}<0\\}$ of $\Xi$ is parametrized by the restriction of $\widetilde{\xi}$
to $\hat{{\rm J}}_{\xi}=(\pi/2,\pi+\arctan(n_{*}))$. The left picture of
Figure 1 reproduces the separatrix curve (in black); the negative part of the
separatrix curve is highlighted in dashed-yellow. The cusp is the red point
and the horizontal inflection point is coloured in green.

######  Definition 4.4.

The (open) _upper and lower domains_ bounded by the separatrix curve $\Xi$ are
denoted by ${\mathcal{M}}_{\pm}$. In Figure 1, the upper domain
$\mathcal{M}_{+}$ is coloured in three orange tones: orange, dark-orange and
light-orange; the lower domain $\mathcal{M}_{-}$ is coloured in two brown
tones: light-brown and brown.

Figure 1. On the left: the separatrix curve (in black), the upper domain
$\mathcal{M}_{+}$ (coloured in three orange tones) and the lower domain
$\mathcal{M}_{-}$ (coloured in two brown tones). On the right: the curve
${\mathtt{C}}$ and its parametrization obtained by intersecting ${\mathtt{C}}$
with lines through the origin.

######  Proposition 4.5.

The polynomial ${\rm P}_{{\bf c}}$ has multiple roots if and only if ${\bf
c}\in\Xi\cup Oy$, has four complex roots if and only if ${\bf
c}\in{\mathcal{M}}_{-}\setminus(Oy\cap{\mathcal{M}}_{-}\,)$, and has three
distinct real roots if and only if ${\bf
c}\in{\mathcal{M}}_{+}\setminus(Oy\cap{\mathcal{M}}_{+}\,)$. Equivalently,

${\mathcal{A}}={\mathcal{M}}_{-}\setminus(Oy\cap{\mathcal{M}}_{-}\,),\quad{\mathcal{B}}={\mathcal{M}}_{+}\setminus(Oy\cap{\mathcal{M}}_{+}\,),\quad{\mathcal{C}}=\Xi\cup
Oy.$

###### Proof.

First, we prove the following claim. Claim. ${\rm P}_{{\bf c}}$ has a double
root $a_{3}\neq 0$ if and only if ${\bf c}$ belongs to the separatrix curve
minus the cusp.

Note that $c_{1}\neq 0$ (otherwise the double root would be $0$). Let $a_{4}$
be the other simple real root and $b_{1}+ib_{2}$, $b_{1}-ib_{2}$, $b_{2}>0$,
be the two complex conjugate roots. Since the sum of the roots of ${\rm
P}_{{\bf c}}$ is zero, we have $b_{1}=-\frac{1}{2}(2a_{3}+a_{4})$. Since the
coefficient of $x^{3}$ is zero and $b_{2}>0$, we get
$b_{2}=\sqrt{2a_{3}^{2}+a_{3}a_{4}+3a_{4}^{2}/4}$. Expanding
$(x-a_{3})^{2}(x-a_{4})(x-b_{1}-ib_{2})(x-b_{1}+ib_{2})$ and comparing the
coefficients of the monomials $x^{n}$, $n=1,\dots,4$, with the coefficients of
${\rm P}_{{\bf c}}$ we may write $c_{1}$ and $c_{2}$ as functions of $a_{3}$
and $a_{4}$,

(4.3)
$\begin{split}c_{1}&=\frac{1}{27}(3a_{3}^{4}+6a_{3}^{3}a_{4}+4a_{3}^{2}a_{4}^{2}+2a_{3}a_{4}^{3}),\\\
c_{2}&=-\frac{2}{3}(4a_{3}^{3}+3a_{3}^{2}a_{4}+2a_{3}a_{4}^{2}+a_{4}^{3}).\end{split}$

In addition,

$c_{1}^{2}=\frac{2}{27}(3a_{3}^{4}a_{4}+2a_{3}^{3}a_{4}^{2}+a_{3}^{2}a_{4}^{3}).$

Taking into account that $a_{3}\neq 0$, it follows that $(a_{3},a_{4})$
belongs to the algebraic curve ${\mathtt{C}}$ (the black curve on the right
picture in Figure 1) defined by the equation

$54y(3x^{2}+2xy+y^{2})-(3x^{3}+6x^{2}y+4xy^{2}+2y^{3})^{2}=0.$

Now, consider the line $\ell_{m,n}$ through the origin, with homogeneous
coordinates $(m,n)$, i.e., the line with parametric equations
$p_{m,n}(t)=(mt,nt)$. If $(m,n)\neq(1,0)$ and
$3m^{3}+6m^{2}n+4mn^{2}+2n^{3}\neq 0$ (we are excluding the two red lines on
the right picture in Figure 1), $\ell_{m,n}$ intersects ${\mathtt{C}}$ when
$t=0$ and $t=t_{m,n}$, where

$t_{m,n}=\frac{3\sqrt[3]{2}\sqrt[3]{n(3m^{2}+2mn+n^{2})}}{\sqrt[3]{(3m^{3}+6m^{2}n+4mn^{2}+2n^{3})^{2}}}.$

If $(m,n)=(1,0)$ or $3m^{3}+6m^{2}n+4mn^{2}+2n^{3}=0$, $\ell_{m,n}$ intersects
${\mathtt{C}}$ only at the origin (see the right picture in Figure 1). Hence
$\beta:[(m,n)]\to t_{m,n}\cdot(m,n)$, $[(m,n)]\neq[(1,0)]$,
$3m^{3}+6m^{2}n+4mn^{2}+2n^{3}\neq 0$, is a parametrization of
${\mathtt{C}}\setminus\\{(0,0)\\}$. Thus, using (4.3), the map

$[(m,n)]\to(c_{1}(\beta([(m,n])),c_{2}(\beta([(m,n]))\in\mathbb{R}^{2}$

is a parametrization of the set of all ${\bf c}$, $c_{1}\neq 0$, such that
$\mathrm{P}_{{\bf c}}$ has multiple roots. It is now a computational matter to
check that $(c_{1}(\beta([(m,n])),c_{2}(\beta([(m,n]))=\xi([(m,n)])$. This
proves the claim. It also shows that ${\rm P}_{{\bf c}}$ has multiple roots if
and only if ${\bf c}\in\Xi\cup Oy$.

To prove the other assertions, we begin by observing that the discriminant of
the derived polynomial ${\rm P}^{\prime}_{{\bf c}}$ is negative. Hence ${\rm
P}^{\prime}_{{\bf c}}$ has two distinct real roots and a pair of complex
conjugate roots. Denote by $x^{\prime}_{{\bf c}}$ and $x^{\prime\prime}_{{\bf
c}}$ the real roots of ${\rm P}^{\prime}_{{\bf c}}$, ordered so that
$x^{\prime}_{{\bf c}}<x^{\prime\prime}_{{\bf c}}$. Observe that
$x^{\prime}_{{\bf c}}$ and $x^{\prime\prime}_{{\bf c}}$ are differentiable
functions of c. Then, ${\rm P}_{{\bf c}}$ possesses three distinct real roots
if and only if $x^{\prime}_{{\bf c}}\cdot x^{\prime\prime}_{{\bf c}}<0$, one
simple real root if and only if $x^{\prime}_{{\bf c}}\cdot
x^{\prime\prime}_{{\bf c}}>0$, and a multiple root if and only if
$x^{\prime}_{{\bf c}}\cdot x^{\prime\prime}_{{\bf c}}=0$.

From the first part of the proof, the set of all ${\bf c}\in\mathbb{R}^{2}$,
such that ${\rm P}_{{\bf c}}$ has only simple roots is the complement of
$\Xi\cup Oy$. This set has five connected components:

(4.4) $\begin{split}{\mathcal{M}}_{+}^{\prime}&=\left\\{{\bf
c}\in{\mathcal{M}}_{+}\setminus(Oy\cap{\mathcal{M}}_{+})\mid
c_{1}<0\right\\},\\\ {\mathcal{M}}_{+}^{\prime\prime}&=\left\\{{\bf
c}\in{\mathcal{M}}_{+}\setminus(Oy\cap{\mathcal{M}}_{+})\mid
c_{1}>0\,\,\text{and}\,\,c_{2}>0\right\\},\\\
{\mathcal{M}}_{+}^{\prime\prime\prime}&=\left\\{{\bf
c}\in{\mathcal{M}}_{+}\setminus(Oy\cap{\mathcal{M}}_{+})\mid
c_{1}>0\,\,\text{and}\,\,c_{2}<0\right\\},\\\
{\mathcal{M}}_{-}^{\prime}&=\left\\{{\bf
c}\in{\mathcal{M}}_{-}\setminus(Oy\cap{\mathcal{M}}_{-})\mid
c_{1}<0\right\\},\\\ {\mathcal{M}}_{-}^{\prime\prime}&=\left\\{{\bf
c}\in{\mathcal{M}}_{-}\setminus(Oy\cap{\mathcal{M}}_{-})\mid
c_{1}>0\right\\}.\end{split}$

Referring to the left picture in Figure 1, ${\mathcal{M}}_{+}^{\prime}$ is the
orange domain, ${\mathcal{M}}_{+}^{\prime\prime}$ is the dark-orange domain,
${\mathcal{M}}_{+}^{\prime\prime\prime}$ is the light-orange domain,
${\mathcal{M}}_{-}^{\prime}$ is the light-brown domain, and
${\mathcal{M}}_{-}^{\prime\prime}$ is the brown domain.

Consider the following points (the black points in Figure 1):

$\begin{split}&{\bf c}_{1}=(-2,1)\in{\mathcal{M}}^{\prime}_{+},\quad{\bf
c}_{2}=(1/6,8)\in{\mathcal{M}}_{+}^{\prime\prime},\quad{\bf
c}_{3}=(1/6,-8)\in{\mathcal{M}}_{+}^{\prime\prime\prime},\\\ &{\bf
c}_{4}=(-6,-9)\in{\mathcal{M}}_{-}^{\prime},\quad{\bf
c}_{5}=(4,-9)\in{\mathcal{M}}_{-}^{\prime\prime}.\end{split}$

Using Klein’s formulas for the icosahedral solution of a quintic polynomial in
principal form (cf. [23, 32, 37]),333We used the Trott and Adamchik code (cf.
[37]) implementing Klein’s formulas in the software Mathematica. we find that
the polynomials ${\rm P}_{{\bf c}_{j}}$, $j=1,2,3$, have three distinct real
roots and that ${\rm P}_{{\bf c}_{j}}$, $j=4,5$, have one real root. The
domain ${\mathcal{M}}^{\prime}_{+}$ is connected and the function
${\mathcal{M}}^{\prime}_{+}\ni{\bf c}\mapsto x^{\prime}_{{\bf c}}\cdot
x^{\prime\prime}_{{\bf c}}$ is differentiable and nowhere zero. Since
$x^{\prime}_{{\bf c}_{1}}\cdot x^{\prime\prime}_{{\bf c}_{1}}<0$, it follows
that ${\bf c}\mapsto x^{\prime}_{{\bf c}}\cdot x^{\prime\prime}_{{\bf c}}$ is
strictly negative. Then, ${\rm P}_{{\bf c}}$ has three distinct real roots,
for every ${\bf c}\in{\mathcal{M}}^{\prime}_{+}$. Similarly, ${\rm P}_{{\bf
c}}$ has three distinct real roots, for every ${\bf
c}\in{\mathcal{M}}^{\prime\prime}_{+}\cup{\mathcal{M}}^{\prime\prime\prime}_{+}$
and a unique real root for every ${\bf
c}\in{\mathcal{M}}^{\prime}_{-}\cup{\mathcal{M}}^{\prime\prime}_{-}$. This
concludes the proof. ∎

Figure 2. On the left: the phase curve of ${\bf c}\in{\mathcal{A}}$. On the
right: the phase curve of ${\bf c}\in{\mathcal{B}}$

######  Remark 4.6.

The real roots of ${\rm P}_{{\bf c}_{1}}$ are
$e_{1}=-2.44175<e_{2}=-0.9904<0<e_{3}=2.87645$ and those of ${\rm P}_{{\bf
c}_{2}}$ are $e_{1}=-2.14118<e_{2}=-0.448099<0<e_{3}=0.0701938$. Instead, the
roots of ${\rm P}_{{\bf c}_{3}}$ are $0<e_{1}=0.12498<e_{2}=0.250656<2.15383$.
Since the product $e_{2}({\bf c})\,e_{3}({\bf c})$ is a continuous function on
the connected components ${\mathcal{M}}_{+}^{\prime}$,
${\mathcal{M}}_{+}^{\prime\prime}$, and
${\mathcal{M}}_{+}^{\prime\prime\prime}$, we deduce that the lowest roots of
${\rm P}_{{\bf c}}$ are negative if ${\bf
c}\in{\mathcal{M}}_{+}^{\prime}\cup{\mathcal{M}}_{+}^{\prime\prime}$ and
positive if ${\bf c}\in{\mathcal{M}}_{+}^{\prime\prime\prime}$.

###  4.2. Phase curves and signatures

######  Definition 4.7.

Let $\Sigma_{{\bf c}}$ be the real algebraic curve defined by $y^{2}={\rm
P}_{\bf c}(x)$. We call $\Sigma_{{\bf c}}$ the _phase curve_ of ${\bf c}$.

If ${\bf c}\in\mathcal{A}\cup\mathcal{B}$, $\Sigma_{{\bf c}}$ is a smooth real
cycle of a hyperelliptic curve of genus $2$. If ${\bf c}\in\mathcal{C}$, and
${\bf c}\neq 0$, $\Sigma_{{\bf c}}$ is a singular real cycle of an elliptic
curve. If ${\bf c}=0$, $\Sigma_{{\bf c}}$ is a singular rational curve. The
following facts can be easily verified:

  * •

if ${\bf c}\in{\mathcal{A}}$, $\Sigma_{{\bf c}}$ is connected, unbounded, and
intersects the $Ox$-axis at $(e_{1},0)$ (see Figure 2);

  * •

if ${\bf c}\in{\mathcal{B}}$, $\Sigma_{{\bf c}}$ has two smooth connected
components, one is compact and the other is unbounded. Let
$\Sigma^{\prime}_{{\bf c}}$ be the compact connected component and
$\Sigma^{\prime\prime}_{{\bf c}}$ be the noncompact one.
$\Sigma^{\prime}_{{\bf c}}$ intersects the $Ox$-axis at $(e_{1},0)$ and
$(e_{2},0)$, while $\Sigma^{\prime\prime}_{{\bf c}}$ intersects the $Ox$-axis
at $(e_{3},0)$ (see Figure 2);

  * •

if ${\bf c}\in{\mathcal{C}}$ and $c_{1}\neq 0$ , $\Sigma_{{\bf c}}$ has a
smooth, unbounded connected component $\Sigma^{\prime\prime}_{{\bf c}}$ and an
isolated singular point $(e_{1},0)$, where $e_{1}=e_{2}$ is the double real
root of ${\rm P}_{\bf c}(x)$. The unbounded connected component intersects the
$Ox$-axis at $(e_{3},0)$, where $e_{3}$ is the simple real root of ${\rm
P}_{\bf c}(x)$ (see Figure 3). If $c_{1}=0$ and $c_{2}\neq 0$, $\Sigma_{{\bf
c}}$ is connected, with an ordinary double point (see Figure 3). If ${\bf
c}=0$, $\Sigma_{{\bf c}}$ is connected with a cusp at the origin (see Figure
3).

######  Definition 4.8.

Let $\gamma$ be a critical curve with nonconstant twist and modulus ${\bf c}$.
Let ${\rm J}_{\gamma}\subset\mathbb{R}$ be the maximal interval of definition
of $\gamma$. With reference to (3.18), we adapt to our context the terminology
used in [7, 24, 28] and call

$\sigma_{\gamma}:{\rm J}_{\gamma}\to\mathbb{R}^{2},\quad
s\mapsto\Big{(}\tau(s),\sqrt{3/2}\,\tau(s){\tau^{\prime}}(s)\Big{)}$

the _signature_ of $\gamma$.

######  Remark 4.9.

From the Poincaré–Bendixson Theorem, it follows that the twist of $\gamma$ is
periodic if and only if $\sigma_{\gamma}({\rm J}_{\gamma})$ is compact.
Observing that $\sigma_{\gamma}({\rm J}_{\gamma})$ is one of the 1-dimensional
connected components of $\Sigma_{{\bf c}}$, we can conclude that the twist is
a periodic function if and only if ${\bf c}\in{\mathcal{B}}$ and
$\sigma_{\gamma}({\rm J}_{\gamma})=\Sigma^{\prime}_{{\bf c}}$.

Figure 3. On the left: the phase curve of ${\bf c}\in{\mathcal{C}}$,
$c_{1}\neq 0$. On the center: the phase curve of ${\bf c}\in{\mathcal{C}}$,
$c_{1}=0$ and $c_{2}\neq 0$. On the right: the phase curve of ${\bf c}=(0,0)$.

######  Definition 4.10.

A critical curve $\gamma$ with modulus ${\bf c}$ is said to be of type
${\mathcal{B}}^{\prime}$ if ${\bf c}\in{\mathcal{B}}$ and
$\sigma_{\gamma}({\rm J}_{\gamma})=\Sigma^{\prime}_{{\bf c}}$; it is said to
be of type ${\mathcal{B}}^{\prime\prime}$ if ${\bf c}\in{\mathcal{B}}$ and
$\sigma_{\gamma}({\rm J}_{\gamma})=\Sigma^{\prime\prime}_{{\bf c}}$.

###  4.3. The twist of a critical curve

####  4.3.1. The twist of a critical curve of type ${\mathcal{A}}$

Let $\gamma$ be a critical curve of type ${\mathcal{A}}$, i.e., with modulus
${\bf c}\in{\mathcal{A}}$. Then ${\rm P}_{{\bf c}}$ has a unique real root
$e_{1}$. The polynomial ${\rm P}_{{\bf c}}(x)$ is positive if $x>e_{1}$ and is
negative if $x<e_{1}$. Since ${\rm P}_{{\bf c}}(0)=-27c_{1}^{2}/2<0$, the root
is positive. Let $\omega_{{\bf c}}>0$ be the improper hyperelliptic integral
of the first kind defined by

$\omega_{{\bf c}}=\sqrt{\frac{3}{2}}\int_{e_{1}}^{+\infty}\frac{\tau
d\tau}{\sqrt{{\rm P}_{{\bf c}}(\tau)}}>0.$

The incomplete hyperelliptic integral

$h_{{\bf c}}(\tau)=\sqrt{\frac{3}{2}}\int_{e_{1}}^{\tau}\frac{udu}{\sqrt{{\rm
P}_{{\bf c}}(u)}},\quad u\geq e_{1}$

is a strictly increasing diffeomorphism of $[e_{1},+\infty)$ onto
$[0,\omega_{{\bf c}})$ (see Figure 4). The twist is the unique even function
$\tau_{{\bf c}}:(-\omega_{{\bf c}},\omega_{{\bf c}})\to\mathbb{R}$, such that
$\tau_{{\bf c}}=h_{{\bf c}}^{-1}$ on $[0,\omega_{{\bf c}})$. The maximal
domain of definition is ${\rm J}_{{\bf c}}=(-\omega_{{\bf c}},\omega_{{\bf
c}})$. $\tau_{{\bf c}}$ is strictly positive, with vertical asymptotes as
$s\to\mp\omega_{{\bf c}}^{\pm}$ (see Figure 4). Note that $\tau_{{\bf c}}$ is
the solution of the Cauchy problem

${\tau}^{\prime\prime}=\tau^{2}-9c_{1}\tau^{-2}(1-c_{1}\tau^{-1}),\quad\tau(0)=e_{1},\quad{\tau^{\prime}}(0)=0.$

Figure 4. On the left: the graph of the function $h_{{\bf c}}$, ${\bf
c}=(1/2,-4.8)\in{\mathcal{A}}$. The red line is the horizontal asymtpote
$y=e_{1}$. On the right: the graph of the twist. The red lines are the
vertical asymptotes $x=\pm\omega_{{\bf c}}$.

####  4.3.2. The twist of a critical curve of type ${\mathcal{B}}^{\prime}$

Let $e_{1}<e_{2}<e_{3}$ be the simple real roots of ${\rm P}_{{\bf c}}$. The
highest root $e_{3}$ is positive. The lower roots $e_{1}$ and $e_{2}$ are
either both negative or both positive and ${\rm P}_{{\bf c}}$ is positive on
$(e_{1},e_{2})$. Let $\omega_{{\bf c}}>0$ be the complete hyperelliptic
integral of the first kind

(4.5) $\omega_{{\bf c}}={\rm
sign}(e_{1})\sqrt{\frac{3}{2}}\int_{e_{2}}^{e_{1}}\frac{\tau d\tau}{\sqrt{{\rm
P}_{{\bf c}}(\tau)}}>0.$

Let $h_{{\bf c}}$ be the incomplete hyperelliptic integrals of the first kind

$h_{{\bf
c}}(\tau)=\left\\{\begin{array}[]{ccc}\displaystyle\sqrt{\frac{3}{2}}\int_{e_{2}}^{\tau}\frac{udu}{\sqrt{{\rm
P}_{{\bf c}}(u)}},&\tau\in[e_{1},e_{2}],&e_{1}<e_{2}<0,\\\
\displaystyle\sqrt{\frac{3}{2}}\int_{e_{1}}^{\tau}\frac{udu}{\sqrt{{\rm
P}_{{\bf c}}(u)}},&\tau\in[e_{1},e_{2}],&0<e_{1}<e_{2}.\end{array}\right.$

The function $h_{{\bf c}}$ is a diffeomorphism of $[e_{1},e_{2}]$ onto
$[0,\omega_{{\bf c}}]$, strictly decreasing if $e_{1}<e_{2}<0$ and strictly
increasing if $0<e_{1}<e_{2}$ (see Figure 5). The twist $\tau_{{\bf c}}$ is
the even periodic function with least period $2\omega_{{\bf c}}$, obtained by
extending periodically the function $\tau(s)=h_{{\bf c}}^{-1}(s)$ defined on
$[0,\omega_{{\bf c}}]$ and on $[-\omega_{{\bf c}},0]$, respectively.

$\bullet$ If $e_{1}<e_{2}<0$, then $\tau_{{\bf c}}$ is strictly negative with
minimum value $e_{1}$ and maximum value $e_{2}$, attained, respectively, at
$s\equiv\omega_{{\bf c}}\mod 2\omega_{{\bf c}}$ and at $s\equiv 0\mod
2\omega_{{\bf c}}$ (see Figure 5).

$\bullet$ If $0<e_{1}<e_{2}$, then $\tau_{{\bf c}}$ is strictly positive, with
minimum value $e_{1}$ and maximum value $e_{2}$, attained, respectively, at
$s\equiv 0\mod 2\omega_{{\bf c}}$ and at $s\equiv\omega_{{\bf c}}\mod
2\omega_{{\bf c}}$. Observe that $\tau_{{\bf c}}$ is the solution of the
Cauchy problem

(4.6)
$\begin{split}(i)\quad&{\tau^{\prime\prime}}=\tau^{2}-9c_{1}\tau^{-2}(1-c_{1}\tau^{-1}),\,\,\tau(0)=e_{2},\,\,{\tau^{\prime}}(0)=0,\,\,{\rm
if}\,\,e_{1}<e_{2}<0,\\\
(ii)\quad&{\tau^{\prime\prime}}=\tau^{2}-9c_{1}\tau^{-2}(1-c_{1}\tau^{-1}),\,\,\tau(0)=e_{1},\,\,{\tau^{\prime}}(0)=0,\,\,{\rm
if}\,\,0<e_{1}<e_{2}.\end{split}$

Figure 5. On the left: the graph of the function $h_{{\bf c}}$ for a critical
curve of type ${\mathcal{B}}^{\prime}$, with modulus ${\bf
c}\approx(-0.828424,-8.349417)\in{\mathcal{B}}$. On the right: the graph of
the twist, an even periodic function with least period $2\omega_{{\bf c}}$.
The lowest roots $e_{1}$ and $e_{2}$ are negative.

####  4.3.3. The twist of a critical curve of type
${\mathcal{B}}^{\prime\prime}$

The twist of a critical curve of type ${\mathcal{B}}^{\prime\prime}$ can be
constructed as in the case of a critical curve of type ${\mathcal{A}}$. More
precisely, let $e_{3}>0$ be the highest real root of ${\rm P}_{{\bf c}}$ and
$\omega_{{\bf c}}$ be the improper hyperelliptic integral of the first kind
given by

$\omega_{{\bf c}}=\sqrt{\frac{3}{2}}\int_{e_{3}}^{+\infty}\frac{\tau
d\tau}{\sqrt{{\rm P}_{{\bf c}}(\tau)}}>0.$

Let $h_{{\bf c}}(\tau)$ be the incomplete hyperelliptic integral

$h_{{\bf c}}(\tau)=\sqrt{\frac{3}{2}}\int_{e_{3}}^{\tau}\frac{udu}{\sqrt{{\rm
P}_{{\bf c}}(u)}},\quad\tau\geq e_{3}.$

Then, $h_{{\bf c}}$ is a strictly increasing diffeomorphism of
$[e_{3},+\infty)$ onto $[0,\omega_{{\bf c}})$. The twist is the unique even
function $\tau_{{\bf c}}:(-\omega_{{\bf c}},\omega_{{\bf c}})\to\mathbb{R}$,
such that $\tau_{{\bf c}}=h_{{\bf c}}^{-1}$ on $[0,\omega_{{\bf c}})$. The
maximal interval of definition of $\tau_{{\bf c}}$ is ${\rm J}_{{\bf
c}}=(-\omega_{{\bf c}},\omega_{{\bf c}})$. The function $\tau_{{\bf c}}$ is
positive, with vertical asymptotes as $s\to\mp\omega_{{\bf c}}^{\pm}$, and is
the solution of the Cauchy problem

${\tau^{\prime\prime}}=\tau^{2}-9c_{1}\tau^{-2}(1-c_{1}\tau^{-1}),\quad\tau(0)=e_{3},\quad{\tau^{\prime}}(0)=0.$

####  4.3.4. The twist of a critical curve of type ${\mathcal{C}}$ with
$c_{1}\neq 0$

The twist of a critical curve of type ${\mathcal{C}}$, with $c_{1}\neq 0$, can
be constructed as for curves of types ${\mathcal{A}}$ or
${\mathcal{B}}^{\prime\prime}$. Let $e_{3}>0$ be simple real root of ${\rm
P}_{{\bf c}}$ and $\omega_{{\bf c}}$ be the improper elliptic integral of the
first kind

$\omega_{{\bf c}}=\sqrt{\frac{3}{2}}\int_{e_{3}}^{+\infty}\frac{\tau
d\tau}{\sqrt{{\rm P}_{{\bf c}}(\tau)}}>0.$

Let $h_{{\bf c}}(\tau)$ be the incomplete elliptic integral

$h_{{\bf c}}(\tau)=\sqrt{\frac{3}{2}}\int_{e_{3}}^{\tau}\frac{udu}{\sqrt{{\rm
P}_{{\bf c}}(u)}},\quad\tau\geq e_{3}.$

Then, $h_{{\bf c}}$ is a strictly increasing diffeomorphism of
$[e_{3},+\infty)$ onto $[0,\omega_{{\bf c}})$. The twist $\tau_{{\bf c}}$ is
the unique even function $\tau_{{\bf c}}:(-\omega_{{\bf c}},\omega_{{\bf
c}})\to\mathbb{R}$, such that $\tau_{{\bf c}}=h_{{\bf c}}^{-1}$ on
$[0,\omega_{{\bf c}})$. The maximal interval of definition of $\tau_{{\bf c}}$
is ${\rm J}_{{\bf c}}=(-\omega_{{\bf c}},\omega_{{\bf c}})$. The twist is
positive, with vertical asymptotes as $s\to\mp\omega_{{\bf c}}^{\pm}$. Note
that $\tau_{{\bf c}}$ is the solution of the Cauchy problem

${\tau^{\prime\prime}}=\tau^{2}-9c_{1}\tau^{-2}(1-c_{1}\tau^{-1}),\quad\tau(0)=e_{3},\quad{\tau^{\prime}}(0)=0.$

####  4.3.5. The twist of a critical curve with $c_{1}=0$

If $c_{1}=0$, the bending vanishes identically and the twist is a solution of
the second order ODE ${\tau^{\prime\prime}}-\tau^{2}=0$. Then,

$\tau(s)=\sqrt[3]{6}\wp(\frac{s+a}{\sqrt[3]{6}}|0,g_{3}),\,\,\,g_{3}=-\sqrt[3]{2/243}\,c_{2},\quad\kappa(s)=0,$

where $a$ is an unessential constant and $\wp(-,g_{2},g_{3})$ is the
Weierstrass function with invariants $g_{2}$, $g_{3}$.

Figure 6. The nine regions ${\mathcal{A}}_{j}$, ${\mathcal{B}}_{j}$ and
${\mathcal{C}}_{j}$, $j=1,2,3$.

###  4.4. Orbit types and the twelve classes of critical curves with
nonconstant twist

The moduli of the critical curves can be classified depending on the
properties of the eigenvalues of the momenta.

######  Definition 4.11.

For $\mathbf{c}=(c_{1},c_{2})\in\mathbb{R}^{2}$, let $\Delta_{1}({\bf
c})=-27\left(32c_{1}^{3}+9(9+c_{2})^{2}\right)$ be the discriminant of the
cubic polynomial ${\rm Q}_{{\bf c}}$ (cf. (4.2)). We say that ${\bf
c}\in\mathbb{R}^{2}$ is:

  1. (1)

of orbit type 1 (in symbols, ${\bf c}\in{\rm OT}_{1}$) if $\Delta_{1}({\bf
c})>0$; the momentum of a critical curve with modulus ${\bf c}\in{\rm OT}_{1}$
has three distinct real eigenvalues:
$\lambda_{1}=-(\lambda_{2}+\lambda_{3})<0<\lambda_{2}<\lambda_{3}$.

  2. (2)

of orbit type 2 (in symbols, ${\bf c}\in{\rm OT}_{2}$) if $\Delta_{1}({\bf
c})<0$; the momentum of a critical curve with modulus ${\bf c}\in{\rm OT}_{2}$
has a real eigenvalue $\lambda_{1}$ and two complex conjugate roots:
$\lambda_{2}$, with positive imaginary part, and
$\lambda_{3}=\overline{\lambda}_{2}$.

  3. (3)

of orbit type 3 (in symbols, ${\bf c}\in{\rm OT}_{3}$) if $\Delta_{1}({\bf
c})=0$; the momentum of a critical curve with modulus ${\bf c}\in{\rm OT}_{3}$
has an eigenvalue with algebraic multiplicity greater than one ($>1$).

Correspondingly, $\mathbb{R}^{2}$ is partitioned into nine regions (see Figure
6):

${\mathcal{A}}_{j}={\mathcal{A}}\cap{\rm
OT}_{j},\quad{\mathcal{B}}_{j}={\mathcal{B}}\cap{\rm
OT}_{j},\quad{\mathcal{C}}_{j}={\mathcal{C}}\cap{\rm OT}_{j},\quad j=1,2,3.$

######  Definition 4.12.

Let $\gamma$ be a critical curve with modulus ${\bf c}$ and $j\in\\{1,2,3\\}$.
We say that $\gamma$ is of type ${\mathcal{A}}_{j}$ if ${\bf
c}\in{\mathcal{A}}_{j}$; of type ${\mathcal{B}}^{\prime}_{j}$ if ${\bf
c}\in{\mathcal{B}}_{j}$ and the image of its signature $\sigma_{\gamma}$ is
compact; of type ${\mathcal{B}}^{\prime\prime}_{j}$ if ${\bf
c}\in{\mathcal{B}}_{j}$ and the image of $\sigma_{\gamma}$ is unbounded; and
of type ${\mathcal{C}}_{j}$ if ${\bf c}\in{\mathcal{C}}_{j}$, $j=1,2,3$.

######  Remark 4.13.

The only critical curves with periodic twist are those of the types
${\mathcal{B}}^{\prime}_{j}$, $j=1,2,3$. Consequently, critical curves of the
other types cannot be closed.

######  Remark 4.14.

${\mathcal{B}}_{1}$ lies in the half-plane $\\{(c_{1},c_{2})\mid c_{1}<0\\}$;
it is bounded below by $\Xi^{\prime}=\\{{\bf c}\in\Xi\mid c_{1}<0\\}$ and
above by $\Delta^{\prime}=\\{{\bf c}\in\mathbb{R}^{2}\mid\Delta_{1}({\bf
c})=0,\,c_{2}>-9\\}$. The curves $\Xi^{\prime}$ and $\Delta^{\prime}$
intersect each other tangentially at ${\bf
c}^{\prime}=(c^{\prime}_{1},c^{\prime}_{2})\approx(-11.339754,63.004420)$ (see
Figure 7). Thus, ${\mathcal{B}}_{1}$ has two connected components:

${\mathcal{B}}_{1}^{-}=\\{{\bf c}\in{\mathcal{B}}_{1}\mid
c_{1}\in(c^{\prime}_{1},-9)\\},\quad{\mathcal{B}}_{1}^{+}=\\{{\bf
c}\in{\mathcal{B}}_{1}\mid c_{1}>c^{\prime}_{1}\\}.$

Referring to Remark 4.3, $\Xi^{\prime}$ is parametrized by the restriction of
$\widetilde{\xi}$ to the interval $\hat{{\rm
J}}_{\xi}=(\pi/2,\pi+\arctan(n_{*}))$. Let $t^{\prime}$ be the point of
$\hat{{\rm J}}_{\xi}$ such that $\widetilde{\xi}(t^{\prime})={\bf
c}^{\prime}$, ($t^{\prime}\approx 2.3008$). Put $\hat{{\rm
J}}_{\xi}^{-}=(\pi/2,t^{\prime})$ and $\hat{{\rm
J}}_{\xi}^{+}=(t^{\prime},\pi+\arctan(n_{*}))$. The restriction of
$\widetilde{\xi}$ to $\hat{{\rm J}}_{\xi}^{-}$ is a parametrization of
$\Xi_{-}=\\{{\bf c}\in\Xi^{\prime}\mid c_{1}\in(c^{\prime}_{1},0)\\}$ and the
restriction to $\hat{{\rm J}}_{\xi}^{+}$ is a parametrization of
$\Xi_{+}=\\{{\bf c}\in\Xi^{\prime}\mid c_{1}<c^{\prime}_{1}\\}$. Consequently,
${\mathcal{B}}_{1}^{\pm}$ are parametrized by

(4.7)
$\psi_{\pm}:K_{\pm}\ni(t,s)\longmapsto\big{(}\widetilde{\xi}(t)-p(t)\big{)}s+p(t),$

where $K_{\pm}$ are the rectangles $\hat{{\rm J}}_{\xi}^{\pm}\times(0,1)$ and

$p(t)=\big{(}\widetilde{\xi}_{1}(t),\frac{1}{3}(4\sqrt{-2\,\widetilde{\xi}_{1}(t)^{3}}-27)\big{)}.$

Figure 7. On the left: the connected component ${\mathcal{B}}_{1}^{-}$ (dark
brown) of ${\mathcal{B}}_{1}$. The point coloured in cyan is the inflection
point of $\Xi$ (the union of the arcs coloured in black and magenta and of the
two points) and the cusp of $\Delta_{1}=0$ (the union of the yellow and red
arcs and of the two points). The point coloured in purple is the point of
tangential contact. On the right: the connected component
${\mathcal{B}}_{1}^{+}$ (dark brow) of ${\mathcal{B}}_{1}$

##  5\. Integrability by quadratures

###  5.1. Integrability by quadratures of general critical curves

######  Definition 5.1.

Let $\Delta_{2}$ be the polynomial

${\Delta_{2}({\bf
c})=9c_{1}^{3}(c_{1}^{3}+216)+6c_{1}^{3}c_{2}(c_{2}+36)+(c_{2}+9)(c_{2}+18)^{3}.}$

A critical curve $\gamma$ with modulus ${\bf c}$ is said to be _general_ if
$\Delta_{1}({\bf c})\Delta_{2}({\bf c})\neq 0$. Since $\Delta_{1}({\bf c})\neq
0$, the momentum ${\mathfrak{M}}_{\gamma}$ of a general critical curve
$\gamma$ has three distinct eigenvalues $\lambda_{1}$, $\lambda_{2}$,
$\lambda_{3}$, sorted as in Definition 4.11.

Let ${\rm J}$ be the maximal interval of definition of the twist (it can be
computed in terms of the modulus). Define ${\bf y}_{j}:{\rm
J}\to\mathbb{C}^{1,2}$, $j=1,2,3$, by

(5.1)
$\mathbf{y}_{j}={{}^{t}\\!\Big{(}}\tau(3-i{\tau^{\prime}})-\lambda_{j}^{2}-3c_{1},\,9-\frac{9c_{1}}{\tau}-\lambda_{j}\tau-3i{\tau^{\prime}},\,i(\tau^{2}-3\lambda_{j})\Big{)}.$

Let ${\rm V}:{\rm J}\to\mathfrak{gl}(3,\mathbb{C})$ be the matrix-valued map
with column vectors ${\bf y}_{1},{\bf y}_{2}$ and ${\bf y}_{3}$. Let ${\rm
D}(z_{1},z_{2},z_{3})$ denote the diagonal matrix with $z_{j}$ as the $j$th
element on the diagonal. Recall that, if $c_{1}\neq 0$, then $\tau$ is nowhere
zero. We can prove the following.

######  Theorem B.

Let $\gamma:\mathrm{J}\to\mathcal{S}$ be a general critical curve. The
functions ${\rm{det}}({\rm V})$ and $\tau^{2}-3\lambda_{j}$, $j=1,2,3$, are
nowhere zero. Let $r_{j}$ be continuous determinations of
$\sqrt{\tau^{2}-3\lambda_{j}}$ and let $\phi_{j}$ be the functions defined
by444If $c_{1}\neq 0$, the denominator of the integrand in nowhere zero and
the $\phi_{j}$ are real-analytic. If $c_{1}=0$, the integrand reduces to
$(3\tau-\lambda_{j}^{2})(3\lambda_{j}-\tau^{2})^{-1}$. Thus, also in this case
the functions $\phi_{j}$ are real-analytic.

(5.2)
$\phi_{j}(s)=\int_{0}^{s}\frac{3c_{1}\lambda_{j}-(4c_{1}+\lambda_{j}^{2})\tau^{2}(u)+3\tau^{3}(u)}{\tau^{2}(u)(3\lambda_{j}-\tau^{2}(u))}du.$

Then, $\gamma$ is congruent to

${\rm J}\ni s\longmapsto\left[{\rm M}\,{\rm
D}(r_{1}e^{i\phi_{1}},r_{2}e^{i\phi_{2}},r_{3}e^{i\phi_{3}})\,{\rm
V}^{-1}\,\mathbf{e}_{1}\right]\in\mathcal{S},$

where ${\rm M}={\rm V}(0)\,{\rm D}(r_{1}(0),r_{2}(0),r_{3}(0))^{-1}$.

###### Proof.

The proof of Theorem B is organized into three lemmas.

######  Lemma B1.

The following statements hold true:

  1. (1)

if the momentum has three distinct real eigenvalues, then
$\pm\sqrt{3\lambda_{2}}$ and $\pm\sqrt{3\lambda_{3}}$ cannot be roots of ${\rm
P}_{{\bf c}}$;

  2. (2)

if the momentum has two complex conjugate eigenvalues and a positive real
eigenvalue $\lambda_{1}$, then $\pm\sqrt{3\lambda_{1}}$ cannot be roots of
${\rm P}_{{\bf c}}$.

###### Proof of Lemma B1.

First, note that the image of the parametrized curve

$\alpha(t)=\left(-t\Big{(}\frac{t^{3}}{3}+\sqrt{3}\Big{)},t^{3}\Big{(}\frac{t^{3}}{3}+2\sqrt{3}\Big{)}-9\right)$

is contained in the zero locus of $\Delta_{2}$. This can be proved by a direct
computation. Secondly, from the expression of $\mathrm{Q}_{\mathbf{c}}$, it
follows that

(5.3)
$\begin{split}c_{1}&=-\frac{1}{6}(\lambda_{2}^{2}+\lambda_{2}\lambda_{3}+\lambda_{3}^{2})=-\frac{1}{6}(\lambda_{1}^{2}+\lambda_{1}\lambda_{2}+\lambda_{2}^{2}),\\\
c_{2}&=\frac{1}{3}(\lambda_{2}^{2}\lambda_{3}+\lambda_{2}\lambda_{3}^{2}-27)=\frac{1}{3}(\lambda_{1}^{2}\lambda_{2}+\lambda_{1}\lambda_{2}^{2}-27).\end{split}$

(1) Suppose that the momentum has three distinct real eigenvalues. By
contradiction, suppose that $\sqrt{3\lambda_{2}}$ is a root of ${\rm P}_{{\bf
c}}$. Then

$\begin{split}0=-\frac{8}{3}{\rm P}_{{\bf
c}}(\sqrt{3\lambda_{2}})&=\lambda_{3}^{4}+2\lambda_{2}\lambda_{3}^{3}-(\lambda_{2}^{2}-12\sqrt{3}\lambda_{2}^{1/2})\lambda_{3}^{2}-2(\lambda_{2}^{3}-6\sqrt{3}\lambda_{2}^{3/2})\lambda_{3}+\\\
&\quad+(\lambda_{2}^{4}-12\sqrt{3}\lambda_{2}^{5/2}+108\lambda_{2}).\end{split}$

Solving this equation with respect to $\lambda_{3}$, taking into account that
$\lambda_{3}>0$, we obtain

$\lambda_{3}=\frac{1}{2}\left(-\lambda_{2}+\sqrt{5\lambda_{2}^{2}-24\sqrt{3\lambda_{2}}}\right).$

Substituting into (5.3), we find

$c_{1}=\sqrt{3\lambda_{2}}-\frac{1}{3}\lambda_{2}^{2},\quad
c_{2}=-9-2\sqrt{3}\lambda_{2}^{3/2}+\frac{1}{3}\lambda_{2}^{3}.$

Then, ${\bf c}=\alpha(-\sqrt{\lambda_{2}})$. This implies that ${\bf c}$
belongs to the zero locus of $\Delta_{2}$, which is a contradiction. By an
analogous argument, we prove that also $-\sqrt{3\lambda_{2}}$ cannot be a root
of ${\rm P}_{{\bf c}}$. By interchanging the role of $\lambda_{2}$ and
$\lambda_{3}$ and arguing as above, it follows that also
$\pm\sqrt{3\lambda_{3}}$ cannot be roots of ${\rm P}_{{\bf c}}$.

(2) Next, suppose that the momentum has two complex conjugate eigenvalues and
a nonnegative real eigenvalue $\lambda_{1}$. Recall that the eigenvalues are
sorted so that the imaginary part of $\lambda_{2}$ is positive. By
contradiction, suppose that $\sqrt{3\lambda_{1}}$ is a root of ${\rm P}_{{\bf
c}}$. Then,

$\begin{split}0=-\frac{8}{3}{\rm P}_{{\bf
c}}(\sqrt{3\lambda_{1}})&=\lambda_{2}^{4}+2\lambda_{1}\lambda_{2}^{3}-(\lambda_{1}^{2}-12\sqrt{3}\lambda_{1}^{1/2})\lambda_{2}^{2}-2(\lambda_{1}^{3}-6\sqrt{3}\lambda_{1}^{3/2})\lambda_{2}+\\\
&\quad+(\lambda_{1}^{4}-12\sqrt{3}\lambda_{1}^{5/2}+108\lambda_{1}).\end{split}$

Solving this equation with respect to $\lambda_{2}$, taking into account that
the imaginary part of $\lambda_{2}$ is positive, we find

$\lambda_{2}=\frac{1}{2}\left(-\lambda_{1}+\sqrt{5\lambda_{1}^{2}-24\sqrt{3\lambda_{1}}}\right).$

Substituting into (5.3) yields ${\bf c}=\alpha(-\sqrt{\lambda_{1}})$. Thus,
${\bf c}$ is a root of $\Delta_{2}$, which is a contradiction. An analogous
argument shows that $-\sqrt{3\lambda_{1}}$ cannot be a root of ${\rm P}_{{\bf
c}}$. This concludes the proof of the lemma. ∎

######  Lemma B2.

${\rm{det}}({\rm V})(s)\neq 0$, for every $s\in{\rm J}_{\gamma}$.

###### Proof of Lemma B2.

Let ${\mathbb{L}}_{j}$ be the 1-dimensional eigenspaces of the momentum
${\mathfrak{M}}_{\gamma}$ relative to the eigenvalues $\lambda_{j}$. Let $L$
be as in (3.3). By Corollary 3.2 of Theorem A, we have
${\mathcal{F}}\,L\,{\mathcal{F}}^{-1}={\mathfrak{M}}$, where ${\mathcal{F}}$
is a Wilczynski frame field along $\gamma$. Then, ${L}(s)$ and
${\mathfrak{M}}$ have the same eigenvalues. Next, consider the line bundles

$\Lambda_{j}=\\{(s,{\bf y})\in{\rm
J}_{\gamma}\times\mathbb{C}^{1,2}\mid{L}(s){\bf y}=\lambda_{j}{\bf y}\\},\quad
j=1,2,3.$

Note that $(s,{\bf y})\in\Lambda_{j}$ if and only if ${\mathcal{F}}(s){\bf
y}\in{\mathbb{L}}_{j}$. Let ${\bf y}_{j}$, $j=1,2,3$, be as in (5.1). A direct
computation shows that ${L}{\bf y}_{j}=\lambda_{j}{\bf y}_{j}$. Thus, ${\bf
y}_{j}$ is a cross section of the eigenbundle $\Lambda_{j}$. Hence,
${\rm{det}}({\rm V})(s)\neq 0$ if and only if ${\bf y}_{j}(s)\neq\vec{0}$, for
every $s$. Case I: The eigenvalues of the momentum are real and distinct. Let
${\rm y}_{j}^{i}$, $i=1,2,3$, denote the components of ${\bf y}_{j}$. Since
$\lambda_{1}$ is negative, it follows from (5.1) that ${\rm y}^{3}_{1}(s)\neq
0$, for every $s$, and hence $\mathbf{y}_{1}(s)\neq\vec{0}$. We prove that
${\bf y}_{2}(s)\neq\vec{0}$. Suppose, by contradiction, that ${\bf
y}_{2}(s_{*})=\vec{0}$, for some $s_{*}\in{\rm J}_{\gamma}$. From ${\rm
y}^{1}_{2}(s_{*})={\rm y}^{2}_{2}(s_{*})=0$, it follows that
$\tau^{\prime}(s_{*})=0$. Hence $e:=\tau(s_{*})$ is a root of ${\rm P}_{{\bf
c}}$. From ${\rm y}^{3}_{2}(s_{*})=0$, it follows that
$e=\pm\sqrt{3\lambda_{2}}$, which contradicts Lemma B1. An analogous argument
leads to the conclusion that ${\bf y}_{3}(s)\neq\vec{0}$, for every $s\in{\rm
J}_{\gamma}$.

Case II: The momentum has a real eigenvalue $\lambda_{1}$ and two complex
conjugate eigenvalues $\lambda_{2}$, $\lambda_{3}$ ($\lambda_{2}$ with
positive imaginary part). Since $\lambda_{2}$ and $\lambda_{3}$ have nonzero
imaginary parts and $\tau$ is real valued, ${\rm y}_{2}^{3}(s)\neq 0$ and
${\rm y}_{3}^{3}(s)\neq 0$, for every $s$. If $\lambda_{1}<0$, then ${\rm
y}_{1}^{3}(s)\neq 0$, for every $s$. If $\lambda_{1}\geq 0$, suppose, by
contradiction, that ${\bf y}_{1}(s_{*})=\vec{0}$. From ${\rm
y}_{1}^{1}(s_{*})={\rm y}^{2}_{1}(s_{*})=0$, we infer that
$\tau^{\prime}(s_{*})=0$. Hence $e=\tau(s_{*})$ is a root of ${\rm P}_{{\bf
c}}$. From ${\rm y}^{3}_{1}(s_{*})=0$, we have $e=\pm\sqrt{3\lambda_{1}}$,
which contradicts Lemma B1. ∎

We are now in a position to conclude the proof. For $j=1,2,3$, let ${\bf
w}_{j}$ be defined by

${\bf w}_{j}={\mathcal{F}}{\bf y}_{j}:{\rm J}_{\gamma}\to\mathbb{C}^{1,2}.$

Then, ${\bf w}_{j}(s)\in{\mathbb{L}}_{j}$ and ${\bf w}_{j}(s)\neq\vec{0}$, for
every $s$. Thus, there exist smooth functions $\Phi_{j}:{\rm
J}_{\gamma}\to\mathbb{C}$, such that ${\bf w}^{\prime}_{j}=\Phi_{j}{\bf
w}_{j}$. From (2.13), we have

(5.4) $\Phi_{j}{\bf y}_{j}={\bf y}^{\prime}_{j}+K{\bf y}_{j},\quad j=1,2,3,$

where

$K=\begin{pmatrix}ic_{1}\tau^{-2}&-i&\tau\\\ 0&-2ic_{1}\tau^{-2}&1\\\
1&0&ic_{1}\tau^{-2}\\\ \end{pmatrix}.$

Then, the third component of ${\bf y}^{\prime}_{j}+K{\bf y}_{j}$ is equal to

$3\tau+3c_{1}\lambda_{j}\tau^{-2}-(\lambda_{j}^{2}+4c_{1})+i\tau{\tau}^{\prime}.$

Hence, using (5.4) we obtain

(5.5)
$\Phi_{j}=-\frac{\tau\tau^{\prime}}{3\lambda_{j}-\tau^{2}}+i\frac{3c_{1}\lambda_{j}-(4c_{1}+\lambda_{j}^{2})\tau^{2}+3\tau^{3}}{\tau^{2}(3\lambda_{j}-\tau^{2})}.$

######  Lemma B3.

The functions $3\lambda_{j}-\tau^{2}$, $j=1,2,3$, are nowhere zero.

###### Proof of Lemma B3.

The statement is obvious if $\lambda_{j}$ is real and negative or complex,
with nonzero imaginary part. If $\lambda_{j}$ is real non-negative, the
smoothness of $\Phi_{j}$ implies that
$\tau\tau^{\prime}(3\lambda_{j}-\tau^{2})^{-1}$ is differentiable. Then
$(3\lambda_{j}-\tau^{2})(s)\neq 0$, for every $s$, such that
$\tau(s)\tau^{\prime}(s)\neq 0$. If $\tau(s)\tau^{\prime}(s)=0$, it follows
that $\tau(s)$ is a root of the polynomial ${\rm P}_{{\bf c}}$. Therefore, by
Lemma B1, we have that $(3\lambda_{j}-\tau^{2})(s)\neq 0$. ∎

From (5.5) we have

$\int_{0}^{s}\Phi_{j}du=\log\left(\sqrt{\tau^{2}-3\lambda_{j}}\right)+i\phi_{j}+b_{j},\quad
j=1,2,3,$

where $b_{j}$ is a constant of integration, $\sqrt{\tau^{2}-3\lambda_{j}}$ is
a continuous determination of the square root of $\tau^{2}-3\lambda_{j}$ and
$\log(\sqrt{\tau^{2}-3\lambda_{j}})$ is a continuous determination of the
logarithm of $\sqrt{\tau^{2}-3\lambda_{j}}$. Since ${\bf
w}^{\prime}_{j}=\Phi_{j}{\bf w}_{j}$, we obtain

${\mathcal{F}}{\bf y}_{j}r_{j}^{-1}e^{-i\phi_{j}}={\bf m}_{j},\quad j=1,2,3,$

where ${\bf m}_{j}$ is a constant vector belonging to the eigenspace
${\mathbb{L}}_{j}$ of ${\mathfrak{M}}$. This implies

(5.6) ${\mathcal{F}}={\rm
M}\mathrm{D}(r_{1}e^{i\phi_{1}},r_{2}e^{i\phi_{2}},r_{2}e^{i\phi_{2}}){\rm
V}^{-1},$

where ${\rm M}$ is an invertible matrix such that ${\rm
M}^{-1}\,{\mathfrak{M}}\,{\rm
M}=\mathrm{D}(\lambda_{1},\lambda_{2},\lambda_{3})$. By possibly replacing
$\gamma$ with a congruent curve, we may suppose that ${\mathcal{F}}(0)=I_{3}$.
Then, since $\phi_{j}(0)=0$, we have ${\rm M}={\rm V}(0){\rm
D}(r_{1}(0),r_{2}(0),r_{3}(0))^{-1}$. This concludes the proof of Theorem B. ∎

###  5.2. Integrability by quadratures of general critical curves of type
${\mathcal{B}}_{1}^{\prime}$

We now specialize the above procedure to the case of general critical curves
of type ${\mathcal{B}}_{1}^{\prime}$ (i.e., general critical curves with
modulus ${\bf c}\in{\mathcal{B}}_{1}$ and with periodic twist). Let
${\mathcal{M}}_{+}^{\prime}$ be as in (4.4). Since ${\mathcal{B}}_{1}$ is
contained in ${\mathcal{M}}_{+}^{\prime}$, the lowest roots $e_{1}$ and
$e_{2}$ of ${\rm P}_{{\bf c}}$ are negative, for every ${\bf
c}\in{\mathcal{B}}_{1}$ (cf. Remark 4.6).

######  Lemma 5.2.

Let $\gamma$ be a general critical curve of type ${\mathcal{B}}_{1}^{\prime}$.
The $\lambda_{1}$-eigenspace of the momentum is spacelike.

###### Proof.

Let ${\bf y}_{j}$ be as in (5.1). Then ${\mathcal{F}}(s){\bf y}_{j}(s)$
belongs to the $\lambda_{j}$-eigenspace of ${\mathfrak{M}}$, for every
$s\in\mathbb{R}$. Using the conservation law
$\frac{3}{2}\tau^{2}(\tau^{\prime})^{2}={\rm P}_{{\bf c}}(\tau)$ (cf. (3.18))
and taking into account that $\lambda_{j}^{3}+6c_{1}\lambda_{j}+3(9+c_{2})=0$,
we compute

$\langle{\mathcal{F}}{\bf y}_{j},{\mathcal{F}}{\bf y}_{j}\rangle=\langle{\bf
y}_{j},{\bf y}_{j}\rangle=3(\tau^{2}-3\lambda_{j})(2c_{1}+\lambda_{j}^{2}).$

Moreover, since $\lambda_{1}=-(\lambda_{2}+\lambda_{3})$ and
$c_{1}=-(\lambda_{2}^{2}+\lambda_{2}\lambda_{3}+\lambda_{3}^{2})/6$, we have

$\begin{split}&2c_{1}+\lambda_{1}^{2}=\frac{1}{3}(2\lambda_{2}+\lambda_{3})(\lambda_{2}+2\lambda_{3})>0,\\\
&2c_{1}+\lambda_{3}^{2}=\frac{1}{3}(\lambda_{3}-\lambda_{2})(\lambda_{2}+2\lambda_{3})>0,\\\
&2c_{1}+\lambda_{2}^{2}=-\frac{1}{3}(\lambda_{3}-\lambda_{2})(2\lambda_{2}+\lambda_{3})<0.\end{split}$

From the fact that $\lambda_{1}<0$, it follows that $\langle{\bf y}_{1},{\bf
y}_{1}\rangle>0$. This proves that the $\lambda_{1}$-eigenspace of the
momentum is spacelike. ∎

######  Definition 5.3.

There are two possible cases: either the $\lambda_{3}$-eigenspace of
${\mathfrak{M}}$ is spacelike, or else is timelike. In the first case, we say
that $\gamma$ is _positively polarized_ , while in the second case, we say
that $\gamma$ is _negatively polarized_.

######  Remark 5.4.

In view of the above lemma, $\gamma$ is positively polarized if and only if
$e_{1}^{2}-3\lambda_{3}>0$ and is negatively polarized if and only if
$e_{2}^{2}-3\lambda_{3}<0$. It is a linear algebra exercise to prove the
existence of ${A}\in{G}$, such that
${A}^{-1}{\mathfrak{M}}{A}={\mathfrak{M}}_{\lambda_{1},\lambda_{2},\lambda_{3}}$,
where

(5.7)
${\mathfrak{M}}_{\lambda_{1},\lambda_{2},\lambda_{3}}=\begin{pmatrix}\frac{1}{2}(\lambda_{2}+\lambda_{3})&0&\frac{\varepsilon\,i}{2}(\lambda_{2}-\lambda_{3})\\\
0&\lambda_{1}&0\\\
-\frac{\varepsilon\,i}{2}(\lambda_{2}-\lambda_{3})&0&\frac{1}{2}(\lambda_{2}+\lambda_{3})\\\
\end{pmatrix},$

where $\varepsilon=\pm 1$ accounts for the polarization of $\gamma$ (see
below). It is clear that any critical curve of type
${\mathcal{B}}^{\prime}_{1}$ is congruent to a critical curve whose momentum
is in the canonical form
${\mathfrak{M}}_{\lambda_{1},\lambda_{2},\lambda_{3}}$.

######  Definition 5.5.

A critical curve of type ${\mathcal{B}}^{\prime}_{1}$ is said to be in a
_standard configuration_ if its momentum is in the canonical form (5.7). Two
standard configurations with the same twist are congruent with respect to the
left action of the maximal compact abelian subgroup ${\mathbb{T}}^{2}=\\{{\rm
A}\in{\rm G}\mid{\rm A}{\bf e}_{2}\wedge{\bf e}_{2}=0\\}$.

Let ${\bf c}\in{\mathcal{B}}_{1}$, such that $\Delta_{1}({\bf
c})\Delta_{2}({\bf c})\neq 0$. Let $e_{1}<e_{2}<e_{3}$ be the real roots of
${\rm P}_{{\bf c}}$ and let
$\lambda_{1}=-(\lambda_{2}+\lambda_{3})<0<\lambda_{2}<\lambda_{3}$ be the
roots of ${\rm Q}_{{\bf c}}$. Let $\tau$ be the periodic function defined as
in the first of the (4.6) and $\phi_{j}$, $j=1,2,3$, be as in (5.2). Let
$\rho_{j}$ be the constants

$\begin{split}\rho_{1}&=\frac{1}{\sqrt{(2\lambda_{2}+\lambda_{3})(\lambda_{2}+2\lambda_{3})}},\\\
\rho_{2}&=\frac{1}{\sqrt{2(\lambda_{3}-\lambda_{2})(2\lambda_{2}+\lambda_{3})}},\\\
\rho_{3}&=\frac{1}{\sqrt{2(\lambda_{3}-\lambda_{2})(\lambda_{2}+2\lambda_{3})}}\end{split}$

and $z_{j}$ be the functions

(5.8)
$\begin{split}&z_{1}=\rho_{1}\sqrt{3(\lambda_{2}+\lambda_{3})+\tau^{2}}\,e^{i\phi_{1}},\\\
&z_{2}=\rho_{2}\sqrt{3\lambda_{2}-\tau^{2}}\,e^{i\phi_{2}},\\\
&z_{3}=\rho_{3}\sqrt{3\lambda_{3}-\tau^{2}}\,e^{i\phi_{3}}.\end{split}$

Let $\varepsilon=-{\rm sign}(e_{2}^{2}-3\lambda_{3})$. We can state the
following.

######  Theorem C.

A general critical curve of type ${\mathcal{B}}^{\prime}_{1}$ with modulus
${\bf c}$ is congruent to

(5.9) $\gamma:\mathbb{R}\ni
s\longmapsto\left[{{}^{t}\\!(}z_{2}+z_{3},\varepsilon iz_{1},-\varepsilon
i(z_{2}-z_{3}))\right]\in\mathcal{S}.$

In addition, $\gamma$ is in a standard configuration.

###### Proof.

Let $\widetilde{\gamma}$ be a critical curve of type
${\mathcal{B}}^{\prime}_{1}$ with modulus ${\bf c}$. Let ${\mathcal{F}}$ be a
Wilczynski frame along $\widetilde{\gamma}$. Suppose $\varepsilon=1$ (i.e.,
$\langle{\bf y}_{3},{\bf y}_{3}\rangle<0$). Let ${\bf u}_{j}$ be the maps
defined by

$\begin{split}{\bf
u}_{1}&=\frac{1}{\sqrt{3}\sqrt{2c_{1}+\lambda_{1}^{2}}\sqrt{\tau^{2}-3\lambda_{1}}}{\bf
y}_{1},\\\ {\bf
u}_{2}&=\frac{1}{\sqrt{3}\sqrt{-(2c_{1}+\lambda_{2}^{2})}\sqrt{\tau^{2}-3\lambda_{2}}}{\bf
y}_{2},\\\ {\bf
u}_{3}&=\frac{1}{\sqrt{3}\sqrt{2c_{1}+\lambda_{3}^{2}}\sqrt{\tau^{2}-3\lambda_{3}}}{\bf
y}_{3}.\end{split}$

Consider the map ${U}=({\bf u}_{3},{\bf u}_{2},{\bf u}_{1}):\mathbb{R}\to{\rm
GL}(3,\mathbb{C})$. From Theorem B and Lemma 5.2, we have

  * •

$\langle{\bf u}_{1},{\bf u}_{1}\rangle=\langle{\bf u}_{2},{\bf
u}_{2}\rangle=-\langle{\bf u}_{3},{\bf u}_{3}\rangle=1$, and $\langle{\bf
u}_{i},{\bf u}_{j}\rangle=0$, for $i\neq j$, that is, ${U}(s)$ is a pseudo-
unitary basis of $\mathbb{C}^{1,2}$, for every $s\in\mathbb{R}$;

  * •

${U}^{-1}{L}{U}=D(\lambda_{3},\lambda_{2},\lambda_{1})$.

Using again Theorem B, we obtain

(5.10)
$\begin{split}{\mathcal{F}}{U}D(e^{-i\phi_{3}},e^{-i\phi_{2}}&,e^{-i\phi_{1}})=\\\
&{\rm
M}D\Big{(}\sqrt{3(2c_{1}+\lambda_{3}^{2})},\sqrt{-3(2c_{1}+\lambda_{2}^{2})},\sqrt{3(2c_{1}+\lambda_{1}^{2})}\Big{)}^{-1},\end{split}$

where the matrix ${\rm M}\in{\rm GL}(3,\mathbb{C})$ diagonalizes the momentum
of $\widetilde{\gamma}$, that is, ${\rm M}^{-1}\widetilde{{\mathfrak{M}}}{\rm
M}$ $=$ $D(\lambda_{3},\lambda_{2},\lambda_{1})$. In particular, the column
vectors of the right hand side of (5.10), denoted by $\widetilde{{\rm B}}$,
constitutes a pseudo-unitary basis. Let $\epsilon$ be the inverse of a cubic
root of ${\widetilde{\rm B}}$. Then, the column vectors of
$\epsilon{\widetilde{\rm B}}$ constitute a unimodular pseudo-unitary basis.
Therefore there exists a unique ${A}\in{G}$, such that
$\epsilon{A}{\widetilde{\rm B}}={\rm B}$, where

(5.11) ${\rm B}=\begin{pmatrix}\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}&0\\\
0&0&i\\\ \frac{i}{\sqrt{2}}&\frac{i}{\sqrt{2}}&0\\\ \end{pmatrix}.$

Then

(5.12) ${A}{\mathcal{F}}=\epsilon^{-1}{\rm
B}D(e^{i\phi_{3}},e^{i\phi_{2}},e^{i\phi_{1}}){U}^{-1}=\epsilon^{-1}{\rm
B}D(e^{i\phi_{3}},e^{i\phi_{2}},e^{i\phi_{1}})D(-1,1,1)\,{{}^{t}\\!{}\bar{U}}{\bf
h}.$

It is now a computational matter to check that the first column vector of the
right hand side of (5.12) is
$\epsilon^{-1}\,{{}^{t}\\!(}z_{2}+z_{3},iz_{1},-i(z_{2}-z_{3}))$. This implies
$\widetilde{\gamma}={A}^{-1}\gamma$ (i.e., $\widetilde{\gamma}$ and $\gamma$
are congruent to each other).

Taking into account that
${U}^{-1}{L}{U}=D(\lambda_{3},\lambda_{2},\lambda_{1})$ and using (5.10), the
momentum of $\widetilde{\gamma}$ is
$\widetilde{{\mathfrak{M}}}=\widetilde{{\rm
B}}D(\lambda_{3},\lambda_{2},\lambda_{1})\widetilde{{\rm B}}^{-1}$. Therefore,
the momentum of $\gamma$ is

${\mathfrak{M}}={A}\widetilde{{\rm
B}}D(\lambda_{3},\lambda_{2},\lambda_{1})\widetilde{{\rm B}}^{-1}{A}^{-1}={\rm
B}D(\lambda_{3},\lambda_{2},\lambda_{1}){\rm
B}^{-1}={\mathfrak{M}}_{\lambda_{1},\lambda_{2},\lambda_{3}}.$

This proves that $\gamma$ is in standard configuration.

If $\varepsilon=-1$ (i.e., $\langle{\bf y}_{3},{\bf y}_{3}\rangle>0$),
considering ${U}=({\bf u}_{2},{\bf u}_{3},{\bf u}_{1})$ and arguing as above,
we get the same conclusion. ∎

######  Remark 5.6.

Theorem C implies that a standard configuration $\gamma$ does not pass through
the pole $[\mathbf{e}_{3}]$ of the Heisenberg projection $\pi_{H}$. Thus
$\check{\gamma}:=\pi_{H}\circ\gamma$ is a transversal curve of
$\mathbb{R}^{3}$, which does not intersect the $Oz$-axis.

######  Remark 5.7.

Breaking the integrands into partial fractions, the integrals

${f_{j}(\tau)}=\sqrt{\frac{3}{2}}\int_{e_{2}}^{\tau}\frac{3c_{1}\lambda_{j}-(4c_{1}+\lambda_{j}^{2})+3\tau^{3}}{\tau(3\lambda_{j}-\tau^{2})\sqrt{{\rm
P}_{{\bf c}}(\tau)}}d\tau,\quad j=1,2,3,$

can be written as linear combinations of standard hyperelliptic integrals of
the first and third kind. Then $\phi_{j}$ is the odd quasi-periodic function
with quasi-period $2\omega$ such that $\phi_{j}(s)=f_{j}[\tau(s)]$. In
practice, we compute $\tau$ and $\phi_{j}$, $j=1,2,3$, by numerically solving
the following system of ODE,

(5.13)
$\begin{split}\tau^{\prime\prime}&=\tau^{2}-9c_{1}\tau^{-2}(1-c_{1}\tau^{-1}),\\\
\phi_{j}^{\prime}&=\frac{3c_{1}\lambda_{j}-(4c_{1}+\lambda_{j}^{2})\tau^{2}+3\tau^{3}}{\tau^{2}(3\lambda_{j}-\tau^{2})},\quad
j=1,2,3,\end{split}$

with initial conditions

(5.14) $\tau(0)=e_{2},\,\,\tau^{\prime}(0)=0,\,\,\phi_{j}(0)=0,\quad j=1,2,3.$

###  5.3. Closing conditions

From Theorem C, it follows that a critical curve of type
${\mathcal{B}}^{\prime}_{1}$ is closed if and only if

${\mathfrak{P}}_{j}=\frac{1}{2\pi}\phi_{j}(2\omega)\in\mathbb{Q},\quad
j=1,2,3.$

On the other hand,

(5.15)
$\frac{1}{2\pi}\phi_{j}(2\omega)=\frac{1}{\pi}\int_{e_{2}}^{e_{1}}\frac{\sqrt{3}(3c_{1}\lambda_{j}-(4c_{1}+\lambda_{j}^{2})\tau^{2}+3\tau^{3})}{\sqrt{2}\tau(3\lambda_{j}-\tau^{2})\sqrt{{\rm
P}_{{\bf c}}(\tau)}}d\tau.$

Thus, $\gamma$ is closed if and only if the complete hyperelliptic integrals
on the right hand side of (5.15) are rational. For a closed critical curve
$\gamma$, we put ${\mathfrak{P}}_{j}=q_{j}=m_{j}/n_{j}$, where $n_{j}>0$ and
$\gcd(m_{j},n_{j})=1$. We call $q_{j}$, the _quantum numbers_ of $\gamma$. By
construction, $e^{i2\pi{\mathfrak{P}}_{1}},e^{i2\pi{\mathfrak{P}}_{2}}$ and
$e^{i2\pi{\mathfrak{P}}_{3}}$ are the eigenvalues of the _monodromy_
${\mathtt{M}}_{\gamma}={\mathcal{F}}(2\omega){\mathcal{F}}(0)^{-1}$ of
$\gamma$. Since ${\rm det}({\mathtt{M}}_{\gamma})=1$, we have

$\sum_{j=1}^{3}{\mathfrak{P}}_{j}\equiv 0,\quad{\rm mod}\,{\mathbb{Z}}.$

Then, $\gamma$ is closed if and only if two among the integrals
${\mathfrak{P}}_{j}$, $j=1,2,3$, are rational.

######  Remark 5.8.

The closing conditions can be rephrased as follows. Consider the even quasi-
periodic functions $\phi_{1}$, $\phi_{3}$. Then, the critical curve is closed
if and only if the jumps $\phi_{j}|_{0}^{2\omega}$, $j=1,3$, are rational.

######  Example 5.9.

We now consider an example, which will be taken up again in the last section.
Choose ${\bf
c}\approx(-0.8284243304411575,-8.349417691746162)\in{\mathcal{B}}_{1}^{-}$.
The real roots of the quintic polynomial are

$e_{1}\approx-0.931924<e_{2}\approx-0.678034<0<e_{3}\approx 2.79051$

and the eigenvalues of the momentum are

$\lambda_{1}\approx-2.40462<0<\lambda_{2}\approx 0.40614<\lambda_{3}\approx
1.99848.$

The half-period of the twist is computed by numerically evaluating the
hyperelliptic integral (4.5). We evaluate $\tau$, $\phi_{1}$, $\phi_{2}$,
$\phi_{3}$ by solving numerically the system (5.13), with initial conditions
(5.14) on the interval $[-4\omega,4\omega]$. Figure 8 reproduces the graphs of
$\phi_{1}$ and $\phi_{3}$ on the interval $[-4\omega,4\omega]$ (the graph of
the twist was depicted in Figure 5). The red point on the $Ox$-axis is
$2\omega$ and the length of the arrows are the jumps
$\phi_{1}|_{0}^{2\omega}$, $\phi_{3}|_{0}^{2\omega}$. In this example,

$\Big{|}-\frac{1}{2\pi}\frac{2}{15}-\phi_{1}|_{0}^{2\omega}\Big{|}=1.6151\cdot
10^{-8},\quad\Big{|}-\frac{1}{2\pi}\frac{10}{21}-\phi_{3}|_{0}^{2\omega}\Big{|}=-4.46887\cdot
10^{-8}.$

So, modulo negligible numerical errors, the corresponding critical curves are
closed, with quantum number $q_{1}=-2/15$ and $q_{3}=-10/21$. In the last
section we will explain how we computed the modulus. A standard configuration
of a curve with modulus ${\bf c}$ is represented in Figure 13.

Figure 8. The graphs of the quasi-periodic functions $\phi_{1}$ (left) and
$\phi_{3}$ (right), for ${\bf
c}\approx(-0.8284243304411575,-8.349417691746162)\in{\mathcal{B}}_{1}^{-}$.

###  5.4. Discrete global invariants of a closed critical curve

Consider a closed general critical curve $\gamma$ of type
${\mathcal{B}}_{1}^{\prime}$, with modulus ${\bf c}$ and quantum numbers
$q_{1}=m_{1}/n_{1}$, $q_{2}=m_{2}/n_{2}$, $q_{3}=m_{3}/n_{3}$,
$q_{1}+q_{2}+q_{3}\equiv 0\mod\mathbb{Z}$. The half-period $\omega$ of the
twist is given by the complete hyperelliptic integral (4.5). Let
${\mathtt{M}}_{\gamma}={\mathcal{F}}(\omega){\mathcal{F}}(0)^{-1}$ be the
monodromy of $\gamma$. The monodromy does not depend on the choice of the
canonical lift. It is a diagonalizable element of ${G}$ with eigenvalues
$e^{2\pi iq_{1}}$, $e^{2\pi iq_{2}}$, and $e^{2\pi iq_{3}}$. Thus,
${\mathtt{M}}_{\gamma}$ has finite order $n=\mathrm{lcm}(n_{1},n_{3})$. The
momentum ${\mathfrak{M}}_{\gamma}$ has three distinct real eigenvalues, so its
stabilizer is a maximal compact abelian subgroup
${\mathbb{T}}^{2}_{\gamma}\cong{\rm S}^{1}\times{\rm S}^{1}$ of ${G}$ (if
$\gamma$ is a standard configuration,
${\mathbb{T}}^{2}_{\gamma}={\mathbb{T}}^{2}$). Since
$[{\mathtt{M}}_{\gamma},{\mathfrak{M}}_{\gamma}]=0$,
${\mathtt{M}}_{\gamma}\in{\mathbb{T}}^{2}_{\gamma}$. Let $s_{1}$, $s_{3}$ be
the integers defined by $n=s_{1}n_{1}=s_{3}n_{3}$. The CR spin of $\gamma$ is
$1/3$ if and only if $n\equiv 0\mod 3$ and $m_{1}s_{1}\equiv
m_{3}s_{3}\not\equiv 0\mod 3$. The wave number $\mathbf{n}_{\gamma}$ of
$\gamma$ is $n$ if the spin is 1 and $n/3$ if the spin is $1/3$. Let
$|[\gamma]|$ denote the trajectory of $\gamma$. The stabilizer
$\hat{{G}}_{\gamma}=\\{[A]\in[{G}]\mid[{A}]\cdot|[\gamma]|=|[\gamma]|\\}$ is
spanned by $[{\mathfrak{M}}_{\gamma}]$ and is a cyclic group of order ${\bf
n}_{\gamma}$. Geometrically, $\hat{{G}}_{\gamma}$ is the _symmetry group_ of
the critical curve $\gamma$. The CR turning number ${\it w}_{\gamma}$ is the
degree of the map $\mathbb{R}/2n\omega\mathbb{Z}$ $\ni$ $s\mapsto
F_{1}-iF_{3}$ $\in$ $\dot{\mathbb{C}}:=\mathbb{C}\setminus\\{0\\}$, where the
$F_{j}$’s are the components of a Wilczynski frame along $\gamma$. Without
loss of generality, we may suppose that $\gamma$ is in a standard
configuration. From (5.9), it follows that ${\it w}_{\gamma}$ is the degree of
$\mathbb{R}/2n\omega\mathbb{Z}\ni s\mapsto z_{3}\in\dot{\mathbb{C}}$, if
$\varepsilon_{\gamma}=1$, and is the degree of
$\mathbb{R}/2n\omega\mathbb{Z}\ni s\mapsto z_{2}\in\dot{\mathbb{C}}$, if
$\varepsilon_{\gamma}=-1$. Therefore,

${\it w}_{\gamma}=\begin{cases}s_{3}m_{3},\quad{\rm
if}\,\varepsilon_{\gamma}=1,\\\ s_{2}m_{2},\quad{\rm
if}\,\,\varepsilon_{\gamma}=-1.\end{cases}$

A closed critical curve $\gamma$ has an additional discrete CR invariant,
denoted by ${\rm tr}_{*}(\gamma)$, the trace of $\gamma$ with respect to the
spacelike $\lambda_{1}$-eigenspace of the momentum. To clarify the geometrical
meaning of the trace, it is convenient to consider a standard configuration.
In this case, ${\mathbb{L}}_{1}$ is spanned by
$\mathbf{e}_{2}\in\mathbb{C}^{1,2}$ and the corresponding chain is the
intersection of $\mathcal{S}$ with the projective line $z_{2}=0$. The
Heisenberg projection of this chain is the upward oriented $Oz$-axis. Thus,
$\mathrm{tr}_{*}(\gamma)$ is the linking number ${\rm
Lk}(\check{\gamma},Oz^{\uparrow})$ of the Heisenberg projection of $\gamma$
with the upward oriented $Oz$-axis.

######  Proposition 5.10.

Let $\gamma$ be as above. Then

${\rm tr}_{*}(\gamma)=\begin{cases}(q_{1}-q_{3}){\bf n}_{\gamma},\quad{\rm
if}\,\varepsilon_{\gamma}=1,\\\ (q_{1}-q_{2}){\bf n}_{\gamma},\quad{\rm
if}\,\,\varepsilon_{\gamma}=-1.\end{cases}$

###### Proof.

Without loss of generality, we may assume that $\gamma$ is in standard
configuration. The Heisenberg projection of $\gamma$ is

$\check{\gamma}={{}^{t}\\!}\left(\mathrm{Re}\Big{(}\frac{\varepsilon
iz_{1},}{z_{2}+z_{3}}\Big{)},{\mathrm{Im}\Big{(}\frac{\varepsilon
iz_{1},}{z_{2}+z_{3}}\Big{)}},\mathrm{Re}\Big{(}\frac{-\varepsilon
i(z_{2}-z_{3})}{z_{2}+z_{3}}\Big{)}\right).$

Since $\check{\gamma}$ does not intersect the $Oz$-axis, the linking number
${\rm Lk}(\check{\gamma},Oz^{\uparrow})$ is the degree of

$\mathbb{R}/2{\bf n}_{\gamma}\omega\mathbb{Z}\ni
s\longmapsto\frac{z_{1}}{z_{2}+z_{3}}\in\dot{\mathbb{C}}.$

From (5.8) it follows that this degree is the degree of

$f:\mathbb{R}/2{\bf n}_{\gamma}\omega\mathbb{Z}\ni
s\longmapsto\frac{\rho_{1}\sqrt{3(\lambda_{2}+\lambda_{3})+\tau^{2}(s)}\,e^{i\phi_{1}}}{\rho_{2}\sqrt{3\lambda_{2}-\tau^{2}(s)}\,e^{i\phi_{2}}+\rho_{3}\sqrt{3\lambda_{3}-\tau^{2}(s)}\,e^{i\phi_{3}}}.$

Suppose that $\gamma$ is negatively polarized. Then,
$\tau^{2}-3\lambda_{3}<\tau^{2}-3\lambda_{2}<0$ and $0<\tau^{2}<3\lambda_{2}$.
Therefore,

$0<\frac{\rho_{2}\sqrt{3\lambda_{2}-\tau^{2}}}{\rho_{3}\sqrt{3\lambda_{3}-\tau^{2}}}=\sqrt{\frac{(3\lambda_{2}-\tau^{2})(\lambda_{2}+2\lambda_{3})}{(3\lambda_{3}-\tau^{2})(2\lambda_{2}+\lambda_{3})}}\leq\sqrt{\frac{\lambda_{2}(\lambda_{2}+2\lambda_{3})}{\lambda_{3}(2\lambda_{2}+\lambda_{3})}}<1.$

Thus

$f=\frac{\rho_{1}\sqrt{3(\lambda_{2}+\lambda_{3})+\tau^{2}}}{\rho_{3}\sqrt{3\lambda_{3}-\tau^{2}}}\frac{e^{i(\phi_{1}-\phi_{3}})}{1+h\,e^{i(\phi_{2}-\phi_{3})}},$

where

$h=\frac{\rho_{2}\sqrt{3\lambda_{2}-\tau^{2}}}{\rho_{3}\sqrt{3\lambda_{3}-\tau^{2}}}.$

Since $0<h<1$, the image of $1+he^{i(\phi_{2}-\phi_{3})}$ is a curve contained
in a disk of radius $<1$ centered at $(1,0)$. Hence
$1+he^{i(\phi_{2}-\phi_{3})}$ is null-homotopic in $\dot{\mathbb{C}}$. This
implies

${\rm deg}(f)=\frac{1}{2\pi}(\phi_{1}-\phi_{3})\Big{|}_{0}^{2{\bf
n}_{\gamma}\omega}={\bf n}_{\gamma}(q_{1}-q_{3}).$

Suppose that $\gamma$ is positively polarized. Then,
$\tau^{2}-3\lambda_{2}>\tau^{2}-3\lambda_{3}>0$. In particular,
$\tau^{2}>3\lambda_{3}>0$ and

$0<\frac{\rho_{3}\sqrt{\tau^{2}-3\lambda_{3}}}{\rho_{2}\sqrt{\tau^{2}-3\lambda_{2}}}=\sqrt{\frac{(\tau^{2}-3\lambda_{3})(2\lambda_{2}+\lambda_{3})}{(\tau^{2}-3\lambda_{2})(\lambda_{2}+2\lambda_{3})}}<\sqrt{\frac{2\lambda_{2}+\lambda_{3}}{\lambda_{2}+2\lambda_{3}}}<1.$

Then

$f=-i\,\frac{\rho_{1}\sqrt{3(\lambda_{2}+\lambda_{3})+\tau^{2}}}{\rho_{2}\sqrt{\tau^{2}-3\lambda_{2}}}\frac{e^{i(\phi_{1}-\phi_{2})}}{1+\tilde{h}\,e^{i(\phi_{3}-\phi_{2})}},$

where

$\tilde{h}=\frac{\rho_{3}\sqrt{\tau^{2}-3\lambda_{3}}}{\rho_{2}\sqrt{\tau^{2}-3\lambda_{2}}}.$

Since $0<\tilde{h}<1$, the image of $1+\tilde{h}e^{i(\phi_{3}-\phi_{2})}$ is a
curve contained in a disk of radius $<1$ centered at $(1,0)$. Hence
$1+\tilde{h}e^{i(\phi_{3}-\phi_{2})}$ is null-homotopic in $\dot{\mathbb{C}}$.
This implies

${\rm deg}(f)=\frac{1}{2\pi}(\phi_{1}-\phi_{2})\Big{|}_{0}^{2{\bf
n}_{\gamma}\omega}={\bf n}_{\gamma}(q_{1}-q_{2}).$

∎

Summarizing: _the quantum numbers of a closed critical curve are determined by
the wave number, the CR spin, the CR turning number, and the trace_.

##  6\. Experimental evidence of the existence of infinite countably many
closed critical curves of type ${\mathcal{B}}^{\prime}_{1}$ and examples

This section is of an experimental nature. We use numerical tools, implemented
in the software Mathematica 13.3, to support the claim that there exist
countably many closed critical curves of type ${\mathcal{B}}_{1}^{\prime}$,
with moduli belonging to the connected component ${\mathcal{B}}_{1}^{-}$ of
${\mathcal{B}}_{1}$ (cf. Remark 4.14). The same reasoning applies, as well, if
the modulus belongs to the other connected component ${\mathcal{B}}_{1}^{+}$
of ${\mathcal{B}}_{1}$. We parametrize ${\mathcal{B}}_{1}^{-}$ by the map
$\psi_{-}:K_{-}\to{\mathcal{B}}_{1}^{-}$, defined in (4.7), where $K_{-}$ is
the rectangle $\hat{{\rm J}}_{\xi}^{-}\times(0,1)$, $\hat{{\rm
J}}_{\xi}^{-}=(\pi/2,2.3008)$. We take ${\bf p}=(p_{1},p_{2})\in K_{-}$ as the
fundamental parameters. The modulus $\mathbf{c}=(c_{1},c_{2})$, the roots
$e_{1}<e_{2}<0<e_{3}$ of the quintic polynomial, and the eigenvalues
$\lambda_{1}=-(\lambda_{2}+\lambda_{3})<0<\lambda_{2}<\lambda_{3}$ of the
momentum are explicit functions of the parameters $(p_{1},p_{2})$. Let
$K_{-}^{*}$ be the open set of the general parameters, that is,

$K_{-}^{*}=\left\\{{\bf p}\in K_{-}\mid\Delta_{1}(\psi_{-}({\bf
p}))\Delta_{2}(\psi_{-}({\bf p}))\neq 0\right\\}.$

The complete hyperelliptic integrals ${\mathfrak{P}}_{j}$ can be evaluated
numerically as functions of ${\bf p}\in K_{-}^{*}$. Consider the real analytic
map
${\mathfrak{P}}=({\mathfrak{P}}_{1},{\mathfrak{P}}_{3}):K_{-}^{*}\to\mathbb{R}^{2}$.555Actually,
${\mathfrak{P}}$ is real-analytic on all $K_{-}$. Instead,
${\mathfrak{P}}_{2}$ has a jump discontinuity at the exceptional locus. Choose
${\bf p}_{*}=(2,1/2)\in K_{-}^{*}$ and plot the graphs of the functions
$f_{11}(p_{1})={\mathfrak{P}}_{1}(p_{1},1/2)$,
$f_{12}(p_{2})={\mathfrak{P}}_{1}(2,p_{2})$,
$f_{31}(p_{1})={\mathfrak{P}}_{3}(p_{1},1/2)$, and
$f_{32}(p_{2})={\mathfrak{P}}_{3}(2,p_{2})$ (see Figures 9 and 10).

Figure 9. On the left: the graph of the function $f_{11}$. On the right: the
graph of the function $f_{12}$.

Figure 10. On the left: the graph of the function $f_{31}$. On the right: the
graph of the function $f_{32}$.

The function $f_{11}$ is strictly increasing, while the other three functions
are strictly decreasing. This implies that ${\mathfrak{P}}$ has maximal rank
at ${\bf p}_{*}$. Thus ${\mathcal{P}}_{-}={\mathfrak{P}}(K_{-})$ is a set with
non empty interior. In particular
${\mathcal{P}}_{-}^{r}:={\mathcal{P}}_{-}\cap{\mathbb{Q}}$ is an infinite
countable set and, for every ${\bf q}=(q_{1},q_{2})\in{\mathcal{P}}_{-}^{r}$,
there exists a closed critical curve of type $\mathcal{B^{\prime}}_{1}^{-}$
with quantum numbers $q_{1}$ and $q_{2}$. Figure 11 reproduces the plot of the
map ${\mathfrak{P}}$, an open convex set. The mesh supports a stronger
conclusion: the map ${\mathfrak{P}}$ is 1-1. Therefore, one can assume that,
for every rational point $(q_{1},q_{3})\in{\mathcal{P}}_{-}$, there exists a
unique congruence class of closed critical curves with quantum numbers $q_{1}$
and $q_{3}$. The construction of a standard configuration of a critical curve
associated to a rational point ${\bf q}\in{\mathcal{P}}_{-}$ can be done in
three steps.

Figure 11. The plot of the map ${\mathfrak{P}}$.

Step 1. Choose a rational point ${\bf
q}=(q_{1},q_{3})=(m_{1}/n_{1},m_{3}/n_{3})\in{\mathcal{P}}_{-}$. To find the
parameter ${\bf p}\in K_{-}$, such that ${\mathfrak{P}}({\bf p})={\bf q}$, we
may procede as follows: plot the level curves
${\mathtt{X}}_{q_{1}}={\mathfrak{P}}_{1}^{-1}(q_{1})$ and
${\mathtt{Y}}_{q_{3}}={\mathfrak{P}}_{3}^{-1}(q_{3})$ and choose a small
rectangle ${\rm R}\subset K_{-}$ containing
${\mathtt{X}}_{q_{1}}\cap{\mathtt{Y}}_{q_{2}}$ (see Figure 12). Then we
minimize numerically the function

$\delta_{{\bf q}}:{\rm R}\ni{\bf p}\longmapsto\sqrt{({\mathfrak{P}}_{1}({\bf
p})-q_{1})^{2}+({\mathfrak{P}}_{3}({\bf p})-q_{3})^{2}}.$

We use the stochastic minimization method named “differential evolution” [36]
implemented in Mathematica.

Figure 12. On the left: the point ${\bf
q}=(-2/15,-10/21)\in{\mathcal{P}}_{-}$. On the right: the level curves
${\mathtt{X}}_{-2/15}$ and ${\mathtt{Y}}_{-10/21}$. The dotted curve is the
exceptional locus. The green and the cyan domains are the two connected
components of $K_{-}^{*}$. The brow rectangle is the one chosen for the
numerical minimization of the function $\delta_{{\bf q}}$.

######  Example 6.1.

Let us revisit Example 5.9. Choose ${\bf
q}=(-2/15,-10/21)\in{\mathcal{P}}_{-}$. The plot of the level curves
${\mathtt{X}}_{q_{1}}$ and ${\mathtt{Y}}_{q_{3}}$ is depicted in Figure 12.
Minimizing $\delta_{{\bf q}}$ on the rectangle ${\rm
R}=[1.83,1.86]\times[0.65,0.75]$ (depicted on the right picture in Figure 12)
we obtain ${\bf p}=(1.84438,0.719473)$ and $\delta_{{\bf q}}({\bf
p})=3.26867\cdot 10^{-9}$. So, up to negligible numerical errors, we may
assume ${\bf p}={\mathfrak{P}}^{-1}({\bf q})$. Computing $\psi_{-}({\bf p})$,
we find the modulus ${\bf c}=(c_{1},c_{2})$ of the curve, where
$c_{1}=-0.828424$ and $c_{2}=-8.349418$. With the modulus at hand, we compute
the lowest real roots of the quintic polynomial,
$e_{1}\approx-0.931924<e_{2}\approx-0.678034$, and the roots of the momentum,
namely $\lambda_{1}\approx-2.40462<0<\lambda_{2}\approx
0.40614<\lambda_{3}\approx 1.99848$.

Step 2. We evaluate numerically the integral (4.5) and we get the half-period
$\omega$ of the twist of the critical curve. In our example $\omega\approx
0.732307$. The next step is to evaluate the twist $\tau$. This can be done by
solving numerically the Cauchy problem (4.6) on the interval $[0,2n\omega]$,
$n=\mathrm{lcm}(n_{1},n_{2})$. The bending is given by
$\kappa=c_{1}/\tau^{2}$. Next, we solve the Frenet type linear system (2.13),
with initial condition ${\mathcal{F}}(0)=I_{3}$. Then,
$\widetilde{\gamma}:[0,2n\omega]\ni s\longmapsto[F_{1}(s)]\in\mathcal{S}$ is a
critical curve with quantum numbers $q_{1}$ and $q_{3}$ and ${\mathcal{F}}$ is
a Wilczynski frame field along $\widetilde{\gamma}$. However,
$\widetilde{\gamma}$ is not in a standard configuration.

Step 3. The last step consists in bulding the standard configuration. The
momentum ${\mathfrak{M}}$ of $\widetilde{\gamma}$ is $L(0)$, where $L$ is as
in (3.3). Taking into account that $\tau(0)=e_{2}$, $\tau^{\prime}(0)=0$, and
that $\kappa(0)=c_{1}/e_{2}^{2}$, we get

${\mathfrak{M}}=\begin{pmatrix}0&3\Big{(}1-\frac{c_{1}}{e_{2}}\Big{)}&2ie_{2}\\\
e_{2}&0&3i\Big{(}1-\frac{c_{1}}{e_{2}}\Big{)}\\\ 3i&-ie_{2}&0\\\
\end{pmatrix}.$

The eigenspace of the highest eigenvalue is timelike (i.e., these critical
curves are negatively polarized). We compute the eigenvectors and we build a
unimodular pseudo-unitary basis ${A}=({A}_{1},{A}_{2},{A}_{3})$, such that
${A}_{1}$ is an eigenvector of $\lambda_{3}$, ${A}_{2}$ is an eigenvector of
$\lambda_{2}$, and ${A}_{3}$ is an eigenvector of $\lambda_{1}$. Let ${\rm B}$
be as in (5.11). Consider ${\rm M}={\rm B}{A}^{-1}\in{G}$. Then, $\gamma={\rm
M}\widetilde{\gamma}$ is a standard configuration of a critical curve with
quantum numbers $q_{1}$ and $q_{3}$.

######  Remark 6.2.

The curve $\gamma$ does not pass through the pole of the Heisenberg projection
$\pi_{H}$. So, $\widehat{\gamma}=\pi_{H}\circ\gamma$ is a closed transversal
curve of $\mathbb{R}^{3}$ which does not intersect the $Oz$-axis and ${\rm
tr}_{*}(\gamma)={\rm Lk}(\widehat{\gamma},Oz)$.

Figure 13. The Heisenberg projection of a standard configuration of a critical
curve with quantum numbers $q_{1}=-2/15$ and $q_{2}=-10/21$. The figure on the
left reproduces the fundamental arc $\gamma([0,2\omega))$ (coloured in
yellow). The curve can be constructed by acting with the monodromy on the
fundamental arc.

######  Example 6.3.

Applying Step 2 and Step 3 to Example 6.1 and computing the Heisenberg
projection, we obtain the transversal curve depicted in Figure 13, a non-
trivial transversal knot. The quantum numbers are $q_{1}=-2/15$ and
$q_{2}=-10/21$. Recalling what has been said about the discrete invariants of
a critical curve (cf. Section 5.4), the spin is $1/3$, the wave number is
$n=35$, the CR turning number is $-50$, and the trace is $12$.

######  Example 6.4.

Figures 14 and 15 reproduce the Heisenberg projections of the standard
configurations of critical curves of type ${\mathcal{B}^{\prime}}_{1}^{-}$
with quantum numbers $(-3/10$, $-9/25)$, $(-1/5,-3/7)$, $(5/49,-4/7)$, and
$(-7/36,-23/54)$, respectively. All of them have spin 1. The first is a
trivial torus knot with wave number $n=50$, ${\rm tr}_{*}=3$, and CR turning
number ${\it w}=-18$; the second example is a nontrivial transversal knot with
$n=35$, ${\rm tr}_{*}=8$, and ${\it w}=-15$. The third example is a “tangled”
transversal curve with $n=49$, ${\rm tr}_{*}=33$, and ${\it w}=-28$. The last
example is a nontrivial transversal torus knot with wave number $108$, ${\rm
tr}_{*}=25$, and ${\it w}=-46$.

Figure 14. On the left: the Heisenberg projection of a standard configuration
of a critical curve of type ${\mathcal{B}^{\prime}}_{1}^{-}$ with quantum
numbers $q_{1}=-3/10$, $q_{2}=-9/25$. On the right: the Heisenberg projection
of a standard configuration of a critical curve of type
${\mathcal{B}^{\prime}}_{1}^{-}$ with quantum numbers $q_{1}=-1/5$,
$q_{2}=-3/7$.

Figure 15. On the left: the Heisenberg projection of a standard configuration
of a critical curve of type ${\mathcal{B}^{\prime}}_{1}^{-}$, with quantum
numbers $q_{1}=5/49$, $q_{2}=-4/7$. On the right: the Heisenberg projection of
a standard configuration of a critical curve of type
${\mathcal{B}^{\prime}}_{1}^{-}$, with quantum numbers $q_{1}=-7/36$,
$q_{2}=-23/54$.

######  Remark 6.5.

It is clear that, being numerical approximations, the parametrizations
obtained with this procedure are only approximately periodic.

## References

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