# An endomorphism on immersed curves in the pillowcase

Christopher M. Herald  Department of Mathematics and Statistics, University of
Nevada, Reno, NV 89557<EMAIL_ADDRESS>and  Paul Kirk  Department of
Mathematics, Indiana University, Bloomington, IN 47405<EMAIL_ADDRESS>
###### Abstract.

We examine the holonomy-perturbed traceless $SU(2)$ character variety of the
trivial four-stranded tangle $\\{p_{1},p_{2},p_{3},p_{4}\\}\times[0,1]$ in
$S^{2}\times[0,1]$ equipped with a strong marking, either an earring or a
bypass. Viewing these marked tangles as endomorphisms in the cobordism
category from the four-punctured sphere to itself, we identify the images of
these endomorphisms in the Weinstein symplectic partial category under the
partially defined holonomy-perturbed traceless character variety functor. We
express these endomorphisms on immersed curves in the pillowcase in terms of
doubling and figure eight operations and prove they have the same image.

###### Key words and phrases:

Pillowcase, holonomy perturbation, flat moduli space, traceless character
variety, Lagrangian correspondence

###### 2010 Mathematics Subject Classification:

Primary 57K18, 57K31, 57R58; Secondary 81T13

CH is supported by a Simons Collaboration Grant for Mathematicians. PK
acknowledges support from the Fourier Institute in Grenoble and the Max Planck
Institute in Bonn, where this work was carried out.

##  1\. Introduction

The operation of introducing a particular strong marking (in the sense of [25,
Section 2.4]), which we called the earring, on the product tangle
$(S^{2}\times I,\\{p_{1},p_{2},p_{3},p_{4}\\}\times I)$ induces an endomorphim
on the set of immersed curves in the traceless $SU(2)$ character variety of a
four-punctured two-sphere. The main result in this article completes the
description, begun in [10], of this endomorphism. We also extend the result to
another strong marking which we call the bypass. These two markings are
illustrated on the left and right in Figure 1. Such markings are used in
instanton gauge theory to ensure that critical points of the Chern-Simons
function are separated from the orbit singularities in its domain.

We interpret the operation of introducing a marking on a link or tangle in a
3-manifold as a surgery operation. First, choose a Conway sphere (an embedded
2-sphere meeting the link of tangle transversely in four points, denoted in
this article by $(S^{2},4)$). Then, remove a tubular neighborhood of the
Conway sphere (such a neighborhood is illustrated in the middle in Figure 2).
Finally, glue in either the earring tangle or the bypass tangle illustrated in
Figure 2 (these tangles have trivalent vertices).

From a TQFT perspective, this semi-local operation replaces the product
tangle, which represents the identity endomorphism of the Conway sphere in the
cobordism category, by the earring or bypass tangle. Our main results identify
the image of this morphism in the Weinstein symplectic partial category [36]
under the (partially defined) holonomy perturbed traceless character variety
functor.

The traceless $SU(2)$ character variety $P$ of the Conway sphere is a
pillowcase, analytically isomorphic to the quotient of a torus by its elliptic
involution. It is homeomorphic to a two-sphere, with four distinguished
orbifold points (called corners) which correspond to the four fixed points of
the elliptic involution. The complement of these four points is a smooth open
variety which we denote by $P^{*}$; it is endowed with its Atiyah-Bott-Goldman
symplectic form.

The boundary of the earring (resp. bypass) tangle is a disjoint union of two
Conway spheres, with character variety a product $P_{0}\times P_{1}$ of two
pillowcases. In the earlier article [10], the $s$-holonomy perturbed traceless
character variety of the earring tangle (denoted
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}$)
is identified, for small non-zero $s$, as a smooth genus three surface.
Furthermore, the restriction map

$u_{s}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\to
P_{0}^{*}\times P_{1}^{*},$

a Lagrangian immersion, is determined up to homology. The tools used to prove
the results of [10] are insufficient to determine what endomorphism is induced
on the set of regular homotopy classes of immersed curves in the pillowcase by
Weinstein composition [36] with $u_{s}$.

This article provide a more precise geometric description of the Lagrangian
immersion $u_{s}$, sufficient to determine its action on regular homotopy
classes of immersed curves in the pillowcase. We also identify the
corresponding Lagrangian immersion $u_{s}^{\prime}$ for the bypass tangle and
prove that the endomorphisms induced by $u_{s}$ and $u_{s}^{\prime}$ are
equivalent.

Our main results are the following. Theorem 5.5 states:

######  Theorem A.

For small nonzero perturbation parameter $s$, the restriction-to-the-boundary
map from the holonomy perturbed traceless character variety
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}$
(resp.
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{\prime}$),
a closed genus three surface, to a product of pillowcases

$u_{s}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\to
P_{0}^{*}\times P_{1}^{*}$

is a Lagrangian bifold immersion into the smooth top stratum with respect to
the Atiyah-Bott-Goldman symplectic structure. The compositions ${\rm
proj}_{0}\circ u_{s}$ and ${\rm proj}_{1}\circ u_{s}$ have the same critical
set, a disjoint union of four circles which separate
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}$
into two four-punctured two-spheres. Under these maps to the factors, the four
critical circles smoothly embed and encircle the four corners of $P_{0}$ and
$P_{1}$ and each complementary four-punctured two-sphere in
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}$
embeds into $P_{0}$ and $P_{1}$. The identical statement holds for
$u_{s}^{\prime}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{\prime}\to
P_{0}^{*}\times P_{1}^{*}$.

In this theorem, a Lagrangian bifold refers to a Lagrangian immersion of a
surface into a product of two symplectic surfaces whose projection to each
factor has only fold singularities. We give a precise definition of this
notion, which may be of independent interest, in Section 4.1.

In preparation for the statement of our second main result, let
$\operatorname{\mathrm{Lag}}(P^{*})$ denote the set of good immersed curves in
$P^{*}$. Also, $[\operatorname{\mathrm{Lag}}(P^{*})]$ denotes the set of
regular homotopy classes and $[g]$ denotes the regular homotopy class of
$g\in\operatorname{\mathrm{Lag}}(P^{*})$. These notions are given precise
definitions in Section 6.3.

In Section 6.4, we define the following operations on regular homotopy
classes. First, $D([g])$ refers to the double of
$[g]\in[\operatorname{\mathrm{Lag}}(P^{*})]$ (Definition 6.6). If $g$ is a
good immersion of an interval, $F8([g])$ denotes the regular homotopy class of
the Figure Eight curve supported by $g$ (Definition 6.7 and Figure 5). The
symplectomorphism $\Psi:P_{0}^{*}\to P_{1}^{*}$, defined in Equation (5.5), is
the induced by the cylindrical identification of the two boundary components
of the tangle.

######  Theorem B.

The Lagrangian immersions
$u_{s}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\to
P_{0}^{*}\times P_{1}^{*}$ induce a well defined function

$\lim_{s\to
0}(u_{s})_{*}:[\operatorname{\mathrm{Lag}}(P_{0}^{*})]\to[\operatorname{\mathrm{Lag}}(P_{1}^{*})],$

determined by the following two assertions:

  1. (1)

If $g:J\to P_{0}^{*}$ is a good immersion of an open interval, then

$\lim_{s\to 0}(u_{s})_{*}([g])=\Psi(F8([g])).$

  2. (2)

If $g:S^{1}\to P_{0}^{*}$ is an immersion, then

$\lim_{s\to 0}(u_{s})_{*}([g])=\Psi(D([g])).$

The identical statement holds for
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{\prime}$.

###### Proof.

Assertion (1) is Theorem 6.15. Assertion (2) is Theorem 6.8. $\square$

Limits are used to streamline notation. An alternative formulation is: for any
$g\in\operatorname{\mathrm{Lag}}(P_{0}^{*})$, there exists an $s(g)>0$ so that
for all $0<|s|<s(g)$, $g$ is Weinstein-composable with $u_{s}$ and satisfies
$[(u_{s})_{*}([g])]=\Psi(F8([g]))$ or $\Psi(D([g]))$, and similarly for
$u_{s}^{\prime}$.

We can now explain precisely why Theorem B strengthens the corresponding
result in [10]. In that article, a family of Lagrangian immersions
$v_{\delta}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\to
P_{0}^{*}\times P_{1}^{*},\delta>0$, was constructed, called the model maps,
which satisfy

  * •

$v_{\delta}$ is $\delta$-homotopic to $u_{s}$,

  * •

$\lim_{\delta\to 0}(v_{\delta})_{*}$ satisfies the conclusion of Theorem B.

All the conclusions in [10, Sections 9–11] concerning $v_{\delta}$ depend only
on its satisfying the conclusions of Theorem B. Hence:

######  Theorem C.

All assertions in [10, Sections 9–11] concerning the model map
$v_{\delta}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\to
P_{0}^{*}\times P_{1}^{*}$ are valid for the map $u_{s}$.

###  1.1. Relation to [10]

This article is self-contained and does not presuppose that the reader is
familiar with the results or notation of [10]; we recall and reprove here
those results we need. However, many of the mathematical ideas explored in
this article were known to all four authors of [10], as highlighted by Theorem
C above.

What is new in the present article can be summarized as follows. First, we
carry out several asymptotic calculations, notably Lemmas 3.2 and 4.4, which
provide enough control over $u_{s}$ to conclude it is a Lagrangian bifold with
embedded fold locus. Second, we use symmetries and one calculation, Lemma
6.12, to determine the effect of the Lagrangian correspondence on all immersed
curves up to regular homotopy, rather than up to homology as in [10]. Third,
all results are extended to the bypass marking, using similarities between the
fundamental groups of the earring and bypass tangle complements, as calculated
in Propositions 2.1 and 2.7.

A quick overview of the results of this article can be obtained by glancing at
Figure 2, which illustrates the tangles in $S^{2}\times I$ whose holonomy
perturbed character variety this article is about, and the figures in Section
7, which illustrate the statement of Theorem B for the example of the
traceless character varieties associated to a tangle decomposition of the
$(3,7)$ torus knot.

###  1.2. Context

A reader wishing to understand the project which this article completes may
want to first read the introduction to [10], then read this article, and
finally read Sections 10 and 11 of [10] with Theorem C of the current article
understood. These sections of [10] discuss the low-dimensional topology
context, the relationship to bounding cochains, and Figure Eight bubbling in
immersed Floer theory [6, 7, 11].

One aim of this article and the earlier articles [18, 20, 22, 13] is to
develop an immersed curve calculus in the pillowcase for Kronheimer-Mrowka
reduced singular instanton homology [25, 26, 27].) A calculus for immersed
curves in once-punctured tori for Heegaard-Floer homology is constructed by
Hanselman-Rasmussen-Watson [17] and in pillowcases for Khovanov homology by
Kotelskiy-Watson-Zibrowius [24]. These approaches leverage the fact that the
Riemann mapping theorem permits a construction of Lagrangian-Floer theory in
surfaces which avoids the use of moduli spaces of $J$-holomorphic polygons
[1]. These approaches further make use of the theorem of Haiden-Katzarkov-
Kontsevich [16] concerning representability, by immersed curves, of objects of
the triangulated envelope of the Fukaya category of a punctured surface.

Our results should be viewed in the context of the Atiyah-Floer conjecture
and, more broadly, Floer field theory [33, 34, 35, 8, 7, 11, 3, 12]. From this
perspective, our paper calculates the value of the character variety Floer
field theory functor on the non-trivial endomorphisms of the Conway sphere in
the cobordism category given by the earring and bypass tangles. None of the
proofs in this article depend mathematically on any analytic or algebraic
aspects of Lagrangian-Floer theory or Floer field theory. Although we borrow
terminology from symplectic geometry, the proofs we provide rely only on
classical differential topology, the geometry of $SU(2)$, and the result that
$u_{s}$ and $u_{s}^{\prime}$ are Lagrangian immersions, whose proof can be
found in [9].

As a study of $SU(2)$ character varieties of three-manifolds, Theorems A and B
echo the results of Klassen’s 1993 article [23] which contains a calculation
of the character variety of the Whitehead link complement. While that article
was not focused on symplectic topology, restriction to the character variety
to the boundary (two disjoint tori) gives a Lagrangian immersion into the
product of two pillowcases, just like our $u_{s},u^{\prime}_{s}$. Boozer’s
article [5], which discusses 1-stranded tangles in solid tori, and Smith’s
article, [31] which examines the tangle addition tangle, whose boundary is a
union of three Conway spheres, also describe holonomy perturbed character
varieties and their boundary restriction maps to symplectic character
varieties of dimension greater than two.

Figure 1. The middle illustrates a three-ball neighborhood of a base point
chosen on a link in a three-manifold. The left picture illustrates the result
of introducing the earring atom to the link at this base point, with the blue
arc Poincaré dual to the second Stiefel-Whitney class of an SO(3) bundle over
the tangle complement. The right illustrates the result of introducing the
bypass atom to the link at this point, and working in the trivial bundle.
Figure 2. The middle illustrates the product tangle $(S^{2},4)\times[0,1]$,
with a base point placed on one strand. The left picture illustrates the
earring tangle and the right illustrates the bypass tangle.

Acknowledgement. We thank G. Cazassus and A. Kotelskiy for their many
contributions to this project.

##  2\. Summary of [10] and extension to the bypass tangle

In this section, we recall and reformulate those calculations and notation of
[10] about the earring tangle needed for the present article and we extend
them to the bypass tangle.

(a) (b)

Figure 3.  The earring and bypass tangles in $S^{2}\times I$, enhanced with
perturbation curves. Each labelled arrow represents an element of the
fundamental group; a loop based at a point above the page is obtained by
concatenating a straight path from the base point to the tail of the arrow,
the arrow, and a straight path from the head of the arrow back to the base
point.

Figure 3.a illustrates the earring tangle, enhanced by the addition of two
embedded perturbation curves. Also illustrated is a collection of based loops
in the complement, $a,b,e,f,c,d,g,h,p,q,$ and $w$. (Figure 3.b is the
corresponding illustration for the bypass tangle, which does not appear in
[10], but is analyzed in this article as well.)

The following proposition, which sharpens Proposition 3.1 [10], provides a
presentation of the fundamental group $\Pi$ of the complement of this enhanced
graph. It can be proved either using Wirtinger’s algorithm [29] or by direct
topological arguments. The loops $c,e,g,d,$ and $w$ are not needed to generate
$\Pi$, but naming them simplifies notation throughout the article. Here, and
throughout this article, the inverse of any group element $g$ is denoted
$\bar{g}$.

######  Proposition 2.1.

The set of loops $\\{a,b,f,p,q,h\\}$ generates $\Pi$, and $\Pi$ has the
presentation

$\Pi=\langle a,b,f,p,q,h~{}|~{}[p,\lambda_{p}],~{}[q,\lambda_{q}]\rangle,$

where $\lambda_{p}=bh\text{ and }\lambda_{q}=f\bar{a}h$ denote the longitudes
of the perturbation curves.

The remaining labelled loops in Figure 3.a are expressed in terms of these as

$c=\bar{q}\bar{p}bpq,~{}e=ba\bar{f},~{}g=\bar{q}eq=\bar{q}ba\bar{f}q,~{}d=\bar{c}gf=\bar{q}\bar{p}\bar{b}pba\bar{f}qf,$

and

$w=[\bar{a}h\bar{q}\bar{p},h].$

The inclusions of the boundary components

${(S^{2},4)}^{i}\subset S^{2}\times
I,~{}i={\operatorname{0}},{\operatorname{1}}$

induce peripheral subgroups

$\Pi^{\operatorname{0}}=\langle
a,b,e,f~{}|~{}ba\bar{f}\bar{e}\rangle,\qquad\Pi^{\operatorname{1}}=\langle
d,c,g,f~{}|~{}cd\bar{f}\bar{g}\rangle,$

with the inclusion homomorphisms
$\Pi^{\operatorname{0}}\to\Pi,~{}\Pi^{\operatorname{1}}\to\Pi$ preserving the
labeling of the generators. $\square$

The geometric explanation for the expression for $w$ as a commutator is that
the boundary of the tubular neighborhood of the earring is a torus punctured
once by the arc $w$, with meridian $h$ and longitude $\bar{a}h\bar{q}\bar{p}$.

###  2.1. Traceless $SU(2)$ representations

Identify the Lie group $SU(2)$ with the three-sphere of unit quaternions. The
traceless elements thereby correspond to the equatorial two-sphere of unit
quaternions with real part zero, which we denote by

${\mathbb{S}_{{\bf i}}}=\\{x\in SU(2)~{}|~{}\operatorname{Re}(x)=0\\},$

since it coincides with the conjugacy class of ${{\bf i}}$.

Evaluation on the generators $\\{a,b,f,h,p,q\\}$ embeds
$\operatorname{Hom}(\Pi,SU(2))$ in $SU(2)^{6}$ as an affine algebraic set.
Similarly, $\operatorname{Hom}(\Pi^{0},SU(2))$ and
$\operatorname{Hom}(\Pi^{1},SU(2))$ are affine algebraic sets. Inclusion of
the boundary defines a restriction map

(2.1)
$\operatorname{Hom}(\Pi,SU(2))\to\operatorname{Hom}(\Pi^{0},SU(2))\times\operatorname{Hom}(\Pi^{1},SU(2))$

which is equivariant with respect to the conjugation action of $SU(2)$.

By definition a meridian of a tangle is any based loop freely homotopic to the
boundary of a small disk meeting the (unenhanced) tangle transversely in a
point.

######  Definition 2.2.

For any $s\in{\mathbb{R}}$, we call a homomorphism $\rho:\Pi\to SU(2)$ an
$s$-perturbed traceless $SU(2)$ representation satisfying the $w_{2}$
condition111 In this article, the notion of holonomy perturbation is given a
one-parameter, knot-theoretical treatment via the choice of perturbation
curves and condition (3) of Definition 2.2. We omit any discussion of the
origin of this definition and its relation to generically perturbing the
Chern-Simons function, which is due to Taubes [32]. The family of
perturbations we use are compatible both with holonomy perturbations in
instanton gauge theory, and with Hamiltonian perturbations in Lagrangian-Floer
theory [21, 22, 9].  if $\rho$ satisfies

(1) $\displaystyle\rho(w)=-1$ ($w_{2}$ condition) (2) $\displaystyle
0=\operatorname{Re}(\rho(z))\text{ for any meridian }z\text{ other than $w$}$
(traceless condition) (3)
$\displaystyle\rho(p)=e^{s\operatorname{Im}(\rho(\lambda_{p}))},~{}\rho(q)=e^{s\operatorname{Im}(\rho(\lambda_{q}))}$
($s$-perturbed)

For fixed $s\in{\mathbb{R}}$, we let
$\widetilde{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}\subset\operatorname{Hom}(\Pi,SU(2))$
denote the subspace of $s$-perturbed traceless representations of $\Pi$
satisfying the $w_{2}$ condition and let
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}$
denote its orbit space under the conjugation action of $SU(2)$:

$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}=\widetilde{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}/_{\rm
conjugation}\subset(SU(2)^{6})/_{\rm conjugation}.$

The space
$\widetilde{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}$
is a real analytic subset of $SU(2)^{6}$, because the perturbation condition
is analytic rather than algebraic; it follows that its orbit space
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}$
is a semi-analytic set [4].

###  2.2. Restriction maps to the peripheral traceless character variety

We define $P_{0}$ to be the space of conjugacy classes of traceless
representations of the first peripheral subgroup $\Pi^{0}$:

$P_{0}=\\{\rho\in\operatorname{Hom}(\Pi^{0},SU(2))~{}|~{}0=\operatorname{Re}(\rho(a))=\operatorname{Re}(\rho(b))=\operatorname{Re}(\rho(e))=\operatorname{Re}(\rho(f))\\}/_{\small\rm
conjugation}.$

The space $P_{0}$ is called a pillowcase. It is a two-dimensional real semi-
algebraic set, homeomorphic to $S^{2}$, with four singular points, which we
call corners (see Section 3.2). Denote by $P^{*}_{0}$ the complement of the
four corner points. This is a smooth four-punctured two-sphere [28] equipped
with the Atiyah-Bott-Goldman symplectic form [2, 14]. The smooth stratum
$P^{*}_{0}$ consists of irreducible (i.e., non-abelian) representations. The
corners correspond to the abelian representations. Define

$P_{1}=\\{\rho\in\operatorname{Hom}(\Pi^{1},SU(2))~{}|~{}0=\operatorname{Re}(\rho(c))=\operatorname{Re}(\rho(d))=\operatorname{Re}(\rho(g))=\operatorname{Re}(\rho(f))\\}/_{\small\rm
conjugation},$

the traceless character variety of the other peripheral subgroup $\Pi_{1}$.

Restriction of homomorphisms to the peripheral subgroups defines a map

(2.2)
$u_{s}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\to
P_{0}\times P_{1},$

whose properties are the central topic of this article.

It is explained in [10] that the space
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{0}$
is not a smooth manifold, but rather a singular space obtained by crushing
four singular fibers of a certain Seifert-fibered three-manifold to points.
The restriction map

$u_{0}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{0}\to
P_{0}\times P_{1}$

is not a Lagrangian immersion, even on top stratum. We summarize some
properties about $u_{s}$ for small non-zero $s$ established in [10] and [9].

######  Theorem 2.3.

For small non-zero $s$,
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}$
is a smooth compact surface of genus three. The restriction map $u_{s}$ has
image in the top smooth stratum $P_{0}^{*}\times P_{1}^{*}$ and is a smooth
Lagrangian immersion with respect to the Atiyah-Bott-Goldman symplectic form.
The critical set of $\operatorname{proj}_{0}\circ
u_{s}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\to
P_{0}^{*}$ is a disjoint union of four smooth circles, which separates
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}$
into two four-punctured two-spheres.

This theorem is reproved in Section 3, for the convenience of the reader, to
establish its extension to the bypass tangle, and to indicate how it follows
from Lemma 3.2.

###  2.3. Coordinates on the pillowcase and its embedding in
${\mathbb{R}}^{3}$

It is useful to identify $P_{0}$ with the quotient of the two-torus
$T=S^{1}\times S^{1}$ by the elliptic involution

(2.3) $\iota(e^{\gamma{{\bf i}}},e^{\theta{{\bf i}}})=(e^{-\gamma{{\bf
i}}},e^{-\theta{{\bf i}}})$

via the map

(2.4) $T\to P_{0},~{}(e^{\gamma{{\bf i}}},e^{\theta{{\bf
i}}})\mapsto\big{(}a\mapsto{{\bf i}},~{}b\mapsto e^{\gamma{{\bf k}}}{{\bf
i}},~{}f\mapsto e^{\theta{{\bf k}}}{{\bf i}},~{}e\mapsto
e^{(\gamma+\theta){{\bf k}}}{{\bf i}}\big{)}.$

We use $(\gamma,\theta)$ to denote either a point in $T$ or else in its
universal cover ${\mathbb{R}}^{2}.$ The image of $(\gamma,\theta)$ in $P_{0}$
is denoted $[\gamma,\theta]$, which equals $[-\gamma,-\theta].$ It is proven
in [9] that the Atiyah-Bott-Goldman symplectic form on $P_{0}^{*}$ is given in
these coordinates by $cd\gamma\wedge d\theta$ for some non-zero constant $c$.

Our standard depiction of the pillowcase uses the $[\gamma,\theta]$
coordinates and views $P_{0}$ as the identification space obtained from the
rectangle $[0,\pi]\times[0,2\pi]$ by folding it along $[0,\pi]\times\\{\pi\\}$
and identifying edges. In illustrations and our discussion we make the
following conventions. The bottom left corner is $[0,0]$, and similarly the
bottom right corner is $[\pi,0]$, the top left corner is $[0,\pi]$ and the top
right corner is $[\pi,\pi]$. These corners are joined by the bottom edge
$\\{[\gamma,0]~{}|~{}\gamma\in[0,\pi]\\}$ as well as the top, left, and right
edges.

An alternative way to view $P_{0}$ is as a real semi-algebraic subset of
${\mathbb{R}}^{3}$. The characters of the three peripheral loops $ba^{-1}$,
$fa^{-1}$ and $bf^{-1}$ embed the pillowcase $P_{0}$ as a singular algebraic
surface in ${\mathbb{R}}^{3}$. Explicitly, if $\rho:\Pi_{0}\to SU(2)$, then

$\operatorname{Re}(\rho(ba^{-1}))=\cos\gamma,~{}\operatorname{Re}(\rho(fa^{-1}))=\cos\theta,~{}\text{
and }\operatorname{Re}(\rho(bf^{-1}))=\cos(\gamma-\theta)$

are the values of these three characters on $\rho$. They define the map

(2.5)
$P_{0}\to{\mathbb{R}}^{3},~{}[\gamma,\theta]\mapsto(\cos\gamma,\cos\theta,\cos(\gamma-\theta)),$

which embeds $P_{0}$ into ${\mathbb{R}}^{3}$, with image the two-dimensional
singular real semi-algebraic set

$\\{x^{2}+y^{2}+z^{2}-2xyz=1,x^{2}\leq 1,y^{2}\leq 1,z^{2}\leq 1\\}.$

In this manifestation of $P_{0}$, the corners are the four singular points
$(1,1,1),(-1,-1,1),(-1,1,-1),$ and $(1,-1,-1)$. The bottom edge corresponds to
the arc $\\{(\cos\gamma,1,\cos\gamma)~{}|~{}0\leq\gamma\leq\pi\\}$. The top
edge corresponds to $\\{(\cos\gamma,-1,\cos\gamma)\\}$, the left edge
corresponds to $\\{(1,\cos\theta,\cos\theta)\\}$ and the right to
$\\{(-1,\cos\theta,-\cos\theta)\\}$. A similar remark applies to the other
peripheral subgroup.

The following lemma, which identifies the restriction map of Equation (2.2),
is an immediate consequence of the definitions.

######  Lemma 2.4.

The composite $\operatorname{proj}_{0}\circ
u_{s}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\to
P_{0}$ is determined by the three characters:

(2.6)
$\rho\mapsto\big{(}\operatorname{Re}(\rho(b\bar{a})),\operatorname{Re}(\rho(f\bar{a})),\operatorname{Re}(\rho(b\bar{f}))\big{)},$

and $\operatorname{proj}_{1}\circ
u_{s}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\to
P_{1}$ is determined by the three characters

(2.7)
$\rho\mapsto\big{(}\operatorname{Re}(\rho(c\bar{d})),\operatorname{Re}(\rho(f\bar{d})),\operatorname{Re}(\rho(c\bar{f}))\big{)}.$

Different analytic models of the pillowcase can have different formal tangent
spaces at the four singular points. In this article, $P_{0}$ is understood as
the traceless $SU(2)$ character variety of $(S^{2},4)$, which has a four-
dimensional formal tangent space at each corner (see e.g. [9, Section 4])
whereas the hypersurface $x^{2}+y^{2}+z^{2}-2xyz=1$ has a three-dimensional
formal tangent space at each singular point. None of the arguments in this
article depend on this distinction.

###  2.4. Elimination of variables and partial gauge fixing

The semi-analytic space
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}$
is described above as the orbit space of an $SU(2)$ action on
$\widetilde{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}$.
Following Section 4 of [10], we introduce a lower-dimensional model for
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}$,
namely as a quotient of a subspace of
$\widetilde{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}$
by an involution. The following is a self-contained description of this slice-
with-involution, for the convenience of the reader.

Extend the elliptic involution $\iota$ on $T$ of Equation (2.3) to
$T\times{\mathbb{S}_{{\bf i}}}$ by the formula

(2.8) $\hat{\iota}(e^{\gamma{{\bf i}}},e^{\theta{{\bf i}}},H)=(e^{-\gamma{{\bf
i}}},e^{-\theta{{\bf i}}},-{{\bf i}}H{{\bf i}}).$

Consider the smooth $s$-dependent embedding

(2.9) $L_{s}:T\times{\mathbb{S}_{{\bf
i}}}\xrightarrow{(a,b,f,h,p,q)}SU(2)^{6},~{}$

given by

(2.10) $L_{s}(e^{\gamma{{\bf i}}},e^{\theta{{\bf i}}},H)=(a,b,f,h,p,q)=({{\bf
i}},e^{\gamma{{\bf k}}}{{\bf i}},e^{\theta{{\bf k}}}{{\bf
i}},H,e^{s\operatorname{Im}(e^{\gamma{{\bf k}}}{{\bf
i}}H)},e^{s\operatorname{Im}(e^{\theta{{\bf k}}}H)}).$

The image of $L_{s}(T\times{\mathbb{S}_{{\bf i}}})\subset SU(2)^{6}$ is a
smooth analytic set. Moreover, since $[p,\lambda_{p}]=1$ and
$[q,\lambda_{q}]=1$ for any point in this image, every point in
$L_{s}(T\times{\mathbb{S}_{{\bf i}}})$ determines an $s$-perturbed traceless
representation of $\Pi$. In addition, any representation satisfying these two
conditions of Definition 2.2 can be conjugated so that $(a,b,f,h,p,q)$ lies in
$L_{s}(T\times{\mathbb{S}_{{\bf i}}})$.

To identify the intersection $L_{s}(T\times{\mathbb{S}_{{\bf
i}}})\cap\widetilde{\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}}_{s}$,
we introduce the following function. Let

(2.11) $G(s,-)=(G_{1}(s,-),G_{2}(s,-)):T\times{\mathbb{S}_{{\bf
i}}}\to{\mathbb{R}}^{2},$

be given by

(2.12) $G_{1}(s,e^{\gamma{{\bf i}}},e^{\theta{{\bf
i}}},H)=\operatorname{Re}\left(h\bar{a}h\bar{q}\bar{p}\right)=\operatorname{Re}(pq\bar{h}a\bar{h})$

and

(2.13) $G_{2}(s,e^{\gamma{{\bf i}}},e^{\theta{{\bf
i}}},H)=\operatorname{Re}\left(\bar{a}h\bar{q}\bar{p}\right)=\operatorname{Re}(pq\bar{h}a),$

using Equation (2.10). For any $s\in{\mathbb{R}}$, denote by

$\widehat{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}=(G(s,-))^{-1}(0,0)\subset
T\times{\mathbb{S}_{{\bf i}}}$

the zero set of $G(s,-)$.

The pair of functions
$F=(F_{2},F_{3})=(\operatorname{Re}(\bar{p}a\bar{f}qf),\operatorname{Re}(h\bar{p}a\bar{f}qf))$
were introduced in Section 4 of [10], and in that article,
$\widehat{\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}}_{s}$
is defined to be the preimage of zero under $F(s,-)$. The following lemma
implies that $G(s,-)$ and $F(s,-)$ have the same zero set.

######  Lemma 2.5.

The components of $L_{s}$ satisfy the relation
$\bar{p}a\bar{f}qf=hpq\bar{h}a,$ and

$G=-F.$

Moreover, the following are equivalent.

  * •

The representation determined by $L_{s}(e^{\gamma{{\bf i}}},e^{\theta{{\bf
i}}},H)$ satisfies the $w_{2}$ condition,

  * •

$G(s,e^{\gamma{{\bf i}}},e^{\theta{{\bf i}}},H)=(0,0)$,

  * •

$[\bar{a}h\bar{q}\bar{p},h]=-1$.

###### Proof.

The commutator of two elements of $SU(2)$ equals $-1$ if and only if the
elements are traceless and perpendicular. Moreover, the inverse of a traceless
element is its negative.

Proposition 2.1 implies that $w=[\bar{a}h\bar{q}\bar{p},h]$. Since
$\operatorname{Re}(h)=0$ this implies that $w=-1$ if and only if
$\operatorname{Re}(\bar{a}h\bar{q}\bar{p})=0=\operatorname{Re}(h\bar{a}h\bar{q}\bar{p})$,
that is, if and only if $G=(0,0)$.

Since $[p,bh]=1$ and $[q,f\bar{a}h]=1$ and
$\operatorname{Re}(h)=0=\operatorname{Re}(b)$ on
$L_{s}(T\times{\mathbb{S}_{{\bf i}}})$, $ph=h\bar{p},$ and

$\bar{p}a\bar{f}qf=\bar{p}a\bar{a}hq\bar{h}a=\bar{p}hq\bar{h}a=hpq\bar{h}a.$

It follows that

$G_{1}=\operatorname{Re}\left(h\bar{a}h\bar{q}\bar{p}\right)=\operatorname{Re}(pq\bar{h}a\bar{h})=-\operatorname{Re}(hpq\bar{h}a)=-\operatorname{Re}(\bar{p}a\bar{f}qf),$

and

$G_{2}=\operatorname{Re}(\bar{a}h\bar{q}\bar{p})=\operatorname{Re}(pq\bar{h}a)=\operatorname{Re}(\bar{h}\bar{p}a\bar{f}qf)=-\operatorname{Re}(h\bar{p}a\bar{f}qf).$

$\square$

The simple proof of the following proposition can be found in [10, Section
4.2].

######  Proposition 2.6.

The involution $\hat{\iota}$ preserves
$\widehat{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}$
and acts freely, and, when $s\neq 0$, the composite

$\widehat{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}\stackrel{{\scriptstyle
L_{s}}}{{\hookrightarrow}}\widetilde{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}\to\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}=\widetilde{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}/SU(2)$

induces an isomorphism of semi-analytic sets
$\widehat{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}/\hat{\iota}\cong\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}.$
$\square$

###  2.5. Extension to the bypass tangle

In this section we provide the counterparts of Propositions 2.1 and 2.6 for
the bypass tangle, enhanced with the perturbation curves indicated in Figure
3.b.

The following variant of Proposition 2.1 is proved similarly. In the
statement, $\Pi^{\prime}$ denotes the fundamental group of the complement of
the enhanced bypass tangle.

######  Proposition 2.7.

The set of loops $\\{a,b,f,p,q,h\\}$ generates $\Pi^{\prime}$, and
$\Pi^{\prime}$ has the presentation

$\Pi^{\prime}=\langle
a,b,f,p,q,h~{}|~{}[p,\lambda_{p}],~{}[q,\lambda_{q}]\rangle,$

where $\lambda_{p}=bh\text{ and }\lambda_{q}=f\bar{a}h$ denote the longitudes
of the perturbation curves. As in the earring case, the remaining labelled
loops in Figure 3.b are expressed in terms of these as

$c=\bar{q}\bar{p}bpq,~{}e=ba\bar{f},~{}g=\bar{q}eq=\bar{q}ba\bar{f}q,~{}d=\bar{c}gf=\bar{q}\bar{p}\bar{b}pba\bar{f}qf.$

The inclusions of the boundary components

${(S^{2},4)}^{i}\subset S^{2}\times
I,~{}i={\operatorname{0}},{\operatorname{1}}$

induce peripheral subgroups

$\Pi^{\operatorname{0}}=\langle
a,b,e,f~{}|~{}ba\bar{f}\bar{e}\rangle,\qquad\Pi^{\operatorname{1}}=\langle
c,d,g,f~{}|~{}cd\bar{f}\bar{g}\rangle,$

with the inclusion homomorphisms
$\Pi^{\operatorname{0}}\to\Pi,~{}\Pi^{\operatorname{1}}\to\Pi$ preserving the
labeling of the generators.

For any $s\in{\mathbb{R}}$, we call a homomorphism
$\rho\in\operatorname{Hom}(\Pi^{\prime},SU(2))$ an $s$-perturbed traceless
$SU(2)$ representation  if $\operatorname{Re}(\rho(z))=0$ for all meridians of
the (unenhanced) bypass tangle, and the perturbation condition

$\rho(p)=e^{s\operatorname{Im}(\rho(\lambda_{p}))},~{}\rho(q)=e^{s\operatorname{Im}(\rho(\lambda_{q}))}$

holds for the meridians of the perturbation curves.

As before, we abuse notation and consider evaluation on the generators
$(a,b,f,p,q,h)$ as an inclusion $\operatorname{Hom}(\Pi^{\prime},SU(2))\subset
SU(2)^{6}$. Then, for fixed $s\in{\mathbb{R}}$,

$\widetilde{\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}}_{s}^{\prime}\subset\operatorname{Hom}(\Pi^{\prime},SU(2))\subset
SU(2)^{6}$

is defined to be the (real analytic) set of $s$-perturbed traceless $SU(2)$
representations of $\Pi^{\prime}$ and

$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{\prime}=\widetilde{\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}}_{s}^{\prime}/SU(2)$

its quotient by conjugation.

The embedding $L_{s}:T\times{\mathbb{S}_{{\bf i}}}\subset SU(2)^{6}$ of
Equation (2.9) meets every orbit in
$\widetilde{\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}}_{s}^{\prime}$.
We introduce the replacement $G^{\prime}$, in the case of the bypass tangle,
for the function $G:T\times{\mathbb{S}_{{\bf i}}}\to{\mathbb{R}}^{2}$ of
Equation (2.11).

Recall that on $T\times{\mathbb{S}_{{\bf i}}}$, the relations
$[\lambda_{p},p]=1$ and $[\lambda_{q},q]=1$ hold. Just as in the proof of
Lemma 2.5, $[p,bh]=1$, $[q,f\bar{a}h]=1$,
$\operatorname{Re}(h)=0=\operatorname{Re}(b)$ on
$L_{s}(T\times{\mathbb{S}_{{\bf i}}})$, $ph=h\bar{p},$ and

$\bar{p}a\bar{f}qf=\bar{p}a\bar{a}hq\bar{h}a=\bar{p}hq\bar{h}a=hpq\bar{h}a.$

This implies that $d$ can be rewritten

$d=\bar{q}\bar{p}\bar{b}pba\bar{f}qf=\bar{q}\bar{p}^{2}\bar{b}ba\bar{f}qf=\bar{q}\bar{p}hpq\bar{h}a.$

Define $G^{\prime}(s,-):T\times{\mathbb{S}_{{\bf i}}}\to{\mathbb{R}}^{2}$ by

(2.14) $\displaystyle G^{\prime}(s,e^{\gamma{{\bf i}}},e^{\theta{{\bf i}}},h)$
$\displaystyle=(G_{1}^{\prime}(s,-),G_{2}^{\prime}(s,-))=(\operatorname{Re}(d),\operatorname{Re}(\bar{h}a))$
$\displaystyle=(\operatorname{Re}(\bar{q}\bar{p}hpq\bar{h}a),\operatorname{Re}(\bar{h}a)).$

Furthermore, define

$\widehat{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}^{\prime}=(G^{\prime}(s,-))^{-1}(0,0)\subset
T\times{\mathbb{S}_{{\bf i}}}$

to be the zero set of $G^{\prime}(s,-)$. The easily verified counterpart of
Proposition 2.6 is:

######  Proposition 2.8.

The involution $\hat{\iota}$ preserves
$\widehat{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}^{\prime}$
and acts freely, and, when $s\neq 0$, the composite

$\widehat{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}^{\prime}\stackrel{{\scriptstyle
L_{s}}}{{\hookrightarrow}}\widetilde{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}^{\prime}\to\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{\prime}=\widetilde{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}^{\prime}/SU(2)$

induces an isomorphism of semi-analytic sets
$\widehat{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}^{\prime}/\hat{\iota}\cong\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{\prime}.$
$\square$

Remark. Although $\Pi$ and $\Pi^{\prime}$ are isomorphic, their $s$-perturbed
traceless character varieties
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}$
and
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{\prime}$
are slightly different. In contrast to the earring tangle, where the arc $w$
represents the obstruction to lifting an $SO(3)$ bundle to an $SU(2)$ bundle,
traceless $s$-perturbed representations on the bypass tangle determine flat
structures on the trivial bundle, and hence the $w_{2}$ condition is absent in
the definition of
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}^{\prime}_{s}$.
The perturbation condition takes the same form (Condition (3) of Definition
2.2) for
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}$
and
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{\prime}$.

In particular, the trivalent vertices have different meaning in the earring
and bypass markings: at a trivalent vertex on the earring, the meridian $w$ is
sent to $-1$ and the meridians of the other two incoming edges are sent to
traceless elements of $SU(2)$ by any representation in
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}$.
In contrast, at a trivalent vertex on the bypass, all three meridians of
incoming edges are sent to traceless elements of $SU(2)$ by any representation
in
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{\prime}$.

The maps $G(s,-)$ and $G^{\prime}(s,-)$, which cut out
$\widehat{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}$
and
$\widehat{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}^{\prime}$
in $T\times{\mathbb{S}_{{\bf i}}}$, are different, but our main results hold
for both $u_{s}$ and $u^{\prime}_{s}$, as a consequence of the calculation
(Lemma 3.2) that their second order expansions in $s$ are similar.

###  2.6. The map
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\to
P_{0}$ induced by restriction to $\Pi_{0}$

Going forward, we use ${\rm proj}$ to denote the projection from a product of
spaces to a factor; which factor will be made clear by the context.

Recall from Proposition 2.1 that the two peripheral subgroups are given by
$\Pi^{\operatorname{0}}=\langle a,b,e,f~{}|~{}ba=ef\rangle$, and
$\Pi^{\operatorname{1}}=\langle c,d,g,f~{}|~{}cd=gf\rangle.$ These map to
$\Pi$ and $\Pi^{\prime}$ preserving labels, i.e., by the identical formulas
for $c,d,e,$ and $g$ in Propositions 2.1 and 2.7.

We have the following commutative diagram:

(2.15)
${\widehat{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}}$${T\times{\mathbb{S}_{{\bf
i}}}}$${T}$${\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}=\widehat{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}/\hat{\iota}}$${P_{0}\times
P_{1}}$${P_{0}=T/\iota}$$\scriptstyle{/\hat{\iota}}$$\scriptstyle{\rm
proj}$$\scriptstyle{/\iota}$$\scriptstyle{u_{s}}$$\scriptstyle{\rm proj}$

as well as an identical diagram obtained by replacing
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s},\widehat{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}$
by
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{\prime},\widehat{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}^{\prime}$.

We summarize the discussion of this section in the following proposition. Set
$\mathbb{S}_{{{\bf i}}}^{*}=\mathbb{S}_{{{\bf i}}}\setminus\\{\pm{{\bf
i}}\\}$. The involution $\hat{\iota}$ acts freely on $T\times\mathbb{S}_{{{\bf
i}}}^{*}$, so that $T\times_{\hat{\iota}}\mathbb{S}_{{{\bf i}}}^{*}$ is a
smooth manifold. (The quotient $T\times_{\hat{\iota}}\mathbb{S}_{{{\bf i}}}$
has eight singular points with neighborhoods cones on $\mathbb{R}P^{3}$.)

######  Proposition 2.9.

For any fixed $s$, the zero sets
$\widehat{\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}}_{s},~{}\widehat{\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}}_{s}^{\prime}$
of the two analytic functions

$G(s,-)=(\operatorname{Re}(pq\bar{h}a\bar{h}),\operatorname{Re}(pq\bar{h}a)),~{}G^{\prime}(s,-)=(\operatorname{Re}(\bar{q}\bar{p}hpq\bar{h}a),\operatorname{Re}(\bar{h}a)),$

are analytic varieties in $T\times{\mathbb{S}_{{\bf i}}}^{*}$. These are
preserved by the involution $\hat{\iota}$, and $L_{s}$ induces isomorphisms
$\widehat{\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}}_{s}/\hat{\iota}\cong\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}$
and
$\widehat{\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}}_{s}^{\prime}/\hat{\iota}\cong\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{\prime}$.
Restriction to the boundary defines maps

$u_{s}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\to
P_{0}\times P_{1}~{}\text{ and
}u^{\prime}_{s}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}^{\prime}_{s}\to
P_{0}\times P_{1}.$

The composites of $u_{s}$, $u_{s}^{\prime}$ with the projection to $P_{0}$ are
induced by the projection $T\times{\mathbb{S}_{{\bf i}}}\to T$.

Remark. The remainder of this article analyzes $u_{s}$ and $u^{\prime}_{s}$.
All theorems which follow are true for both
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}$
and
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{\prime}$.
In all cases, the calculations and proofs are simpler for
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{\prime}$
than
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}$.
The reason is that that $G_{2}^{\prime}$ is a simpler function than $G_{2}$.

##  3\. Expansions in $s$ and the identification of
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s},\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{\prime}$
as genus three surfaces.

The main result of this section is Lemma 3.2, which provides asymptotic
expansions in $s$ of the functions $G$ and $G^{\prime}$ of Proposition 2.9.
These are then used, similarly to the argument of [10], to show that $(0,0)$
is a regular value of $G(s,-)$ and $G^{\prime}(s,-)$ for $0<|s|$ sufficiently
small, so that
$\widehat{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}$
and
$\widehat{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}^{\prime}$
are smooth surfaces. We show that
$\widehat{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}$
and
$\widehat{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}^{\prime}$
are compact genus five surfaces on which the involution $\hat{\iota}$ acts
freely, with quotients
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}$
and
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{\prime}$
genus three surfaces.

###  3.1. Coordinates on $T\times{\mathbb{S}_{{\bf i}}}$

We set notational conventions and introduce coordinates $s,\nu,\tau$. We refer
the reader to Equation (2.10) where the $s$-dependent embedding $L_{s}:T\times
C_{{{\bf i}}}\hookrightarrow SU(2)^{6}$, with coordinates $(a,b,f,h,p,q)$, is
defined.

First, to simplify formulas, we write

$(\gamma,\theta)\in({\mathbb{R}}/2\pi{\mathbb{Z}})^{2}\mbox{ for points
}(e^{\gamma{{\bf i}}},e^{\theta{{\bf i}}})\in T.$

Second, observe that when the perturbation parameter $s$ equals zero,
$G=G^{\prime}=(0,\operatorname{Re}(\bar{h}a))$. Hence there exists an
$s_{0}>0$ so that for any perturbation parameter $s\in[-s_{0},s_{0}]$,
$G_{1}(s,e^{\gamma{{\bf i}}},e^{\theta{{\bf i}}},h)=0$ and
$G^{\prime}_{2}(s,e^{\gamma{{\bf i}}},e^{\theta{{\bf i}}},h)=0$ each imply
that $|\operatorname{Re}(h{{\bf i}})|<\tfrac{1}{2}$. Thus, we introduce
coordinates
$(\nu,\tau)\in[-\tfrac{1}{2},\tfrac{1}{2}]\times({\mathbb{R}}/2\pi{\mathbb{Z}})$
on this annulus in ${\mathbb{S}_{{\bf i}}}$, and set

$h(\nu,\tau)=\nu{{\bf i}}+\sqrt{1-\nu^{2}}e^{\tau{{\bf i}}}{{\bf j}}.$

Consider the compact domain

(3.1)
$\mathcal{D}=[-s_{0},s_{0}]\times({\mathbb{R}}/2\pi{\mathbb{Z}})^{2}\times[-\tfrac{1}{2},\tfrac{1}{2}]\times({\mathbb{R}}/2\pi{\mathbb{Z}}).$

and define $\bar{L}:\mathcal{D}\to SU(2)^{6}$ by

(3.2) $\bar{L}(s,\gamma,\theta,\nu,\tau)=(a,b,f,h,p,q)=({{\bf
i}},e^{\gamma{{\bf k}}}{{\bf i}},e^{\theta{{\bf k}}}{{\bf i}},\nu{{\bf
i}}+\sqrt{1-\nu^{2}}e^{\tau{{\bf i}}}{{\bf
j}},e^{s\operatorname{Im}(e^{\gamma{{\bf k}}}{{\bf
i}}h)},e^{s\operatorname{Im}(e^{\theta{{\bf k}}}h)}).$

By construction, the image of $\bar{L}(s,-)$ lies in
$L_{s}(T\times{\mathbb{S}_{{\bf i}}})$ and contains
$\widehat{\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}}_{s}$,
$\widehat{\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}}_{s}^{\prime}$
for $|s|<s_{0}$.

The involution $\hat{\iota}$ corresponds in these coordinates to

(3.3) $(s,\gamma,\theta,\nu,\tau)\mapsto(s,-\gamma,-\theta,\nu,\tau+\pi).$

We denote equivalence classes by $[s,\gamma,\theta,\nu,\tau]$.

Remark. In the remainder of this article, some assertions may hold for
$|s|<s_{0}$ only after $s_{0}$ has been shrunk further, and correspondingly
restricting the domain $\mathcal{D}$. We opt to use the phrase “after
shrinking $s_{0}$ if needed,” rather than introducing new constants.

###  3.2. The maps $\pi_{0}$ and $\pi_{1}$

Define

(3.4) $\pi_{0}:\mathcal{D}/\hat{\iota}\to
P_{0},~{}\pi_{0}([s,\gamma,\theta,\nu,\tau])=[\gamma,\theta].$

Equivalently, using Lemma 2.4, this map can be written as

$\pi_{0}([s,\gamma,\theta,\nu,\tau])=\big{(}\operatorname{Re}(\rho(b\bar{a})),\operatorname{Re}(\rho(f\bar{a})),\operatorname{Re}(\rho(b\bar{f}))\big{)}=(\cos\gamma,\cos\theta,\cos(\gamma-\theta))\in
P_{0}\subset{\mathbb{R}}^{3}.$

Define

(3.5)
$\pi_{1}:\mathcal{D}/\hat{\iota}\to{\mathbb{R}}^{3},~{}\pi_{1}([s,\gamma,\theta,\nu,\tau])=\big{(}\operatorname{Re}(\rho(c\bar{d})),\operatorname{Re}(\rho(f\bar{d})),\operatorname{Re}(\rho(c\bar{f}))\big{)}$

where $c=\bar{q}\bar{p}bpq$ and $d=\bar{q}\bar{p}\bar{b}pba\bar{f}qf$ express
$c,d$ in terms of the components of $\bar{L}$. By Lemma 2.4, the restriction
of $\pi_{1}$ to either
$\widehat{\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}}_{s}$
or
$\widehat{\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}}^{\prime}_{s}$
takes values in the pillowcase

$\\{x^{2}+y^{2}+z^{2}-2xyz=1,x^{2}\leq 1,y^{2}\leq 1,z^{2}\leq
1\\}\subset{\mathbb{R}}^{3}.$

The proof of the following lemma is clear.

######  Lemma 3.1.

The map $u_{s}$ equals the composite

$u_{s}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}=\\{s\\}\times\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\subset\mathcal{D}/\hat{\iota}\xrightarrow{\pi_{0}\times\pi_{1}}P_{0}\times
P_{1}.$

The identical statement holds for $u_{s}^{\prime}$ and
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{\prime}$.
$\square$

In what follows, we abuse notation slightly write

$u_{s}=\pi_{0}\times\pi_{1}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\to
P_{0}\times P_{1}$

or simply $u_{s}=\pi_{0}\times\pi_{1}$ when its domain
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\subset\mathcal{D}/\hat{\iota},$
is understood.

###  3.3. Expansions in $s$

The functions $G_{1},G_{1}^{\prime}:\mathcal{D}\to{\mathbb{R}}$ vanish when
$s=0$ and hence there exist analytic functions
$\widehat{G}_{1},~{}\widehat{G}_{1}^{\prime}:\mathcal{D}\to{\mathbb{R}}$
satisfying

$G_{1}=s\widehat{G}_{1}\text{ and }G_{1}^{\prime}=s\widehat{G}_{1}^{\prime}.$

An important observation for the sequel is that for fixed non-zero $s$, the
zero locus
$\widehat{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}=\\{G(s,-)=0\\}$
coincides with the zero locus of $(\widehat{G}_{1}(s,-),G_{2}(s,-)),$ and
similarly
$\widehat{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}^{\prime}$
coincides with the zero locus of
$(\widehat{G}_{1}^{\prime}(s,-),G_{2}^{\prime}(s,-)).$

######  Lemma 3.2.

After shrinking $s_{0}$ if needed, there exist analytic real valued functions
$R_{i}$, $i=1,\dots,6$ on $\mathcal{D}$ so that for all $|s|\leq s_{0}$,

$\displaystyle\widehat{G}_{1}$
$\displaystyle=-(\sin\gamma\sin\tau-\sin\theta\cos\tau)+2s\cos\gamma\cos\theta+s^{2}R_{1}+G_{2}R_{2},$
$\displaystyle G_{2}$
$\displaystyle=(1-sR_{6})\big{(}\nu-s\cos\gamma+s^{2}R_{3}\big{)},$
$\displaystyle\widehat{G}_{1}^{\prime}$
$\displaystyle=2(-(\sin\gamma\sin\tau-\sin\theta\cos\tau)+2s\cos\gamma\cos\theta)+s^{2}R_{4}+G_{2}^{\prime}R_{5},\text{
and }$ $\displaystyle G_{2}^{\prime}$ $\displaystyle=\nu.$

###### Proof.

Given analytic functions $\phi_{1},\cdots,\phi_{n}$ on $\mathcal{D}$, we use
the shorthand $O(\phi_{1},\cdots,\phi_{n})$ to denote the ideal in the ring of
analytic functions on $\mathcal{D}$ generated by the
$\phi_{1},\cdots,\phi_{n}$.

Since $p=1+s\operatorname{Im}(bh)+s^{2}(\operatorname{Im}(bh))^{2}+O(s^{3})$
and
$q=1+s\operatorname{Im}(f\bar{a}h)+s^{2}(\operatorname{Im}(f\bar{a}h))^{2}+O(s^{3})$,
it follows that

$\operatorname{Re}(pq)=1+O(s^{2})$

and

$\operatorname{Im}(pq)=s(\operatorname{Im}(bh+f\bar{a}h))+s^{2}\operatorname{Im}(\operatorname{Im}(bh)\operatorname{Im}(f\bar{a}h))+O(s^{3}).$

Then,

$\displaystyle G_{2}=\operatorname{Re}(pq\bar{h}a)$
$\displaystyle=\operatorname{Re}(pq)\operatorname{Re}(\bar{h}a)-\operatorname{Re}(\operatorname{Im}(pq)ha)$
$\displaystyle=-\operatorname{Re}(ha)-s\operatorname{Re}((\operatorname{Im}(bh+f\bar{a}h)ha)+O(s^{2})$
$\displaystyle=-\operatorname{Re}(ha)-s\operatorname{Re}((bh+f\bar{a}h)ha)+s\operatorname{Re}(bh+f\bar{a}h)\operatorname{Re}(ha)+O(s^{2})$
$\displaystyle=-\operatorname{Re}(ha)(1-s\operatorname{Re}(bh+f\bar{a}h))-s\operatorname{Re}((bh+f\bar{a}h)ha)+O(s^{2})$

Set $R_{6}=\operatorname{Re}(bh+f\bar{a}h):\mathcal{D}\to{\mathbb{R}}$. Since
$\operatorname{Re}(ha)=-\nu$ and

$\operatorname{Re}((bh+f\bar{a}h)ha)=\operatorname{Re}(bh^{2}a-f)=-\cos\gamma,$
$G_{2}=\nu(1-sR_{6})-s\cos\gamma+O(s^{2})=(1-sR_{6})(\nu-s\cos\gamma)+O(s^{2}).$

For $s$ small enough, $1-sR_{6}$ is non-zero on $\mathcal{D}$, and hence
invertible. Therefore, shrinking $s_{0}$ if needed,

$G_{2}=(1-sR_{6})(\nu-s\cos\gamma+O(s^{2})),$

establishing the second assertion and showing that

$\nu=s\cos\gamma+O(G_{2},s^{2}).$

This implies that $\sqrt{1-\nu^{2}}=1+O(G_{2},s^{2})$ and therefore the
component function $h$ of Equation (3.2), viewed as a quaternion-valued
function $h:\mathcal{D}\to\mathbb{H}$, is given by

(3.6) $h=s\cos\gamma{{\bf i}}+e^{\tau{{\bf i}}}{{\bf j}}+O(G_{2},s^{2}).$

It easily follows that

$sh{{\bf i}}h=s{{\bf i}}-2s^{2}\cos\gamma e^{\tau{{\bf i}}}{{\bf
j}}+O(G_{2},s^{3}).$

Hence

$\displaystyle G_{1}$ $\displaystyle=\operatorname{Re}(pqh{{\bf
i}}h)=\operatorname{Re}(\operatorname{Im}(pq)h{{\bf i}}h)$
$\displaystyle=s\operatorname{Re}((\operatorname{Im}(bh)+\operatorname{Im}(f\bar{a}h))~{}h{{\bf
i}}h)+s^{2}\operatorname{Re}(\operatorname{Im}(bh)\operatorname{Im}(f\bar{a}h)h{{\bf
i}}h)+O(s^{3})$
$\displaystyle=s\operatorname{Re}((\operatorname{Im}(bh)+\operatorname{Im}(f\bar{a}h)){{\bf
i}})$
$\displaystyle\qquad-2s^{2}\cos\gamma\operatorname{Re}((\operatorname{Im}(bh)+\operatorname{Im}(f\bar{a}h))e^{\tau{{\bf
i}}}{{\bf j}})$
$\displaystyle\qquad\qquad+s^{2}\operatorname{Re}(\operatorname{Im}(bh)\operatorname{Im}(f\bar{a}h){{\bf
i}})+O(G_{2},s^{3})$
$\displaystyle=\big{(}-s(\sin\gamma\sin\tau-\sin\theta\cos\tau)-s^{2}\cos\gamma\cos\theta\big{)}$
$\displaystyle\quad-2s^{2}\cos\gamma(-\cos\theta)$
$\displaystyle\qquad+s^{2}\big{(}\cos\gamma\cos\theta\big{)}+O(G_{2},s^{3})$
$\displaystyle=-s(\sin\gamma\sin\tau-\sin\theta\cos\tau)+2s^{2}\cos\gamma\cos\theta+O(G_{2},s^{3})$

The penultimate equality is a straightforward calculation of each of the three
terms, replacing each occurrence of $h$ by the right side of Equation (3.6).
Dividing by $s$ yields the first assertion.

We turn now to the third and fourth assertions. In the coordinates on
$\mathcal{D}$,

$G_{2}^{\prime}=\operatorname{Re}(h\bar{a})=\nu,$

and therefore $h=e^{\tau{{\bf i}}}{{\bf j}}+O(G^{\prime}_{2})$. Since
$\operatorname{Im}(bh)$ is perpendicular to $h$,
$hp=he^{s\operatorname{Im}(bh)}=\bar{p}h$. Similarly, calculation yields that
$qha=ha\bar{q}+O(G_{2}^{\prime})$. Therefore,

$\displaystyle G_{1}^{\prime}$
$\displaystyle=\operatorname{Re}(\bar{q}\bar{p}hpq\bar{h}a)$
$\displaystyle=-\operatorname{Re}((\bar{p}hp)(qha\bar{q}))$
$\displaystyle=-\operatorname{Re}(\bar{p}^{2}h^{2}a\bar{q}^{2})+O(G_{2}^{\prime})$
$\displaystyle=\operatorname{Re}(\bar{q}^{2}\bar{p}^{2}a)+O(G_{2}^{\prime})$
$\displaystyle=2s\big{(}\operatorname{Re}(bha)+\operatorname{Re}(f\bar{a}ha)\big{)}-4s^{2}\operatorname{Re}(\operatorname{Im}(f\bar{a}h)\operatorname{Im}(bh)a)+O(s^{3},G_{2}^{\prime})$
$\displaystyle=-2s(\sin\gamma\sin\tau-\sin\theta\cos\tau)$
$\displaystyle\qquad+4s^{2}\operatorname{Re}(\operatorname{Im}((\cos\theta+\sin\theta{{\bf
k}})ha)\operatorname{Im}((\cos\gamma{{\bf i}}+\sin\gamma{{\bf
j}})h)a)+O(s^{3},G_{2}^{\prime})$
$\displaystyle=-2s(\sin\gamma\sin\tau-\sin\theta\cos\tau)+4s^{2}\cos\theta\cos\gamma+O(s^{3},G_{2}^{\prime}).$

Dividing by $s$ yields the desired calculation of $\widehat{G}_{1}^{\prime}$.

$\square$

To provide parallel arguments that
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}$
and
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{\prime}$
are closed genus three surfaces, we replace $G$ and $G^{\prime}$ with
functions $\bf G$ and ${\bf G}^{\prime}$, respectively, which have the same
zero sets as $G,G^{\prime}$, but with the convenient property that $\bf G$ and
${\bf G}^{\prime}$ certain Taylor expansions in $s$ agree. Define

(3.7) $\displaystyle{\bf G}(s,\gamma,\theta,\nu,\tau)$
$\displaystyle=(\widehat{G}_{1}-G_{2}R_{2},G_{2}(1+sR_{6}))$
$\displaystyle=\big{(}\sin\theta\cos\tau-\sin\gamma\sin\tau+2s\cos\gamma\cos\theta+s^{2}R_{1},\nu-s\cos\gamma+s^{2}R_{3}\big{)}$

and

(3.8) $\displaystyle{\bf G}^{\prime}(s,\gamma,\theta,\nu,\tau)$
$\displaystyle=(\tfrac{1}{2}\widehat{G}_{1}^{\prime}-\tfrac{1}{2}G_{2}^{\prime}R_{5},G_{2}^{\prime})$
$\displaystyle=\big{(}\sin\theta\cos\tau-\sin\gamma\sin\tau+2s\cos\gamma\cos\theta+s^{2}\tfrac{R_{4}}{2},\nu\big{)}$

For any fixed $s$, we’ll let

${\bf G}_{s}={\bf G}(s,-),~{}{\bf G}^{\prime}_{s}={\bf G}^{\prime}(s,-)$

and set

$V_{s}={\bf G}_{s}^{-1}(0,0),~{}V^{\prime}_{s}=({\bf
G}_{s}^{\prime})^{-1}(0,0).$

By construction,

(3.9)
$V_{s}=\widehat{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}\text{
and
}V^{\prime}_{s}=\widehat{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}^{\prime}_{s}\text{
for all non-zero }s.$

Setting $s=0$ in Equations (3.7) and (3.8), one sees that

(3.10) ${\bf
G}_{0}(\gamma,\theta,\tau,\nu)=(\sin\theta\cos\tau-\sin\gamma\sin\tau,\nu)={\bf
G}^{\prime}_{0}(\gamma,\theta,\tau,\nu).$

The following corollary overlaps with Theorem 6.1 from [10] and extends the
result to
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{\prime}$.
In preparation for its statement, define

(3.11) $\mathbb{V}=\\{(s,v)~{}|~{}|s|\leq s_{0},~{}v\in V_{s}\\}={\bf
G}^{-1}(0,0)\subset[-s_{0},s_{0}]\times T\times\mathbb{S}_{{{\bf i}}}^{*}.$

and define $\mathbb{V}^{\prime}$ similarly. The involution $\hat{\iota}$ acts
freely on $[-s_{0},s_{0}]\times T\times\mathbb{S}_{{{\bf i}}}^{*}$, preserving
$\mathbb{V}$.

######  Corollary 3.3.

The composite map

(3.12) $V_{0}\hookrightarrow T\times{\mathbb{S}_{{\bf i}}}\xrightarrow{{\rm
proj}}T$

forms a trivial two-fold covering space over $T^{*}=T\setminus\\{\text{fixed
points}\\}$. The preimage of each fixed point is a smooth circle, and the
union of these four circles is the critical set of the map (3.12). The zero
locus $V_{0}=V_{0}^{\prime}={\bf G}_{0}^{-1}(0,0)$ is a smooth, closed,
connected, orientable surface of genus five.

After shrinking $s_{0}$ if needed, for all $0<|s|\leq s_{0}$,
$\widehat{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}$
and
$\widehat{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}^{\prime}$
are smooth, closed, connected, orientable surfaces of genus five, and
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s},\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{\prime}$
are orientable surfaces of genus three, and the projection
$\operatorname{proj}:\mathbb{V}/\hat{\iota}\to[-s_{0},s_{0}]$ is a smooth
trivial genus three surface bundle with fiber $V_{0}/\hat{\iota}$ over $0$ and
$V_{s}/\hat{\iota}=\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}$
for $s\neq 0$.

###### Proof.

The differential of ${\bf G}_{0}$ is

(3.13) $d{\bf
G}_{0}=\begin{pmatrix}-\cos\gamma\sin\tau&\cos\theta\cos\tau&-\left(\sin\gamma\cos\tau+\sin\theta\sin\tau\right)&0\\\
0&0&0&1\end{pmatrix}.$

If ${\bf G}_{0}(\gamma,\theta,\tau,\nu)=0,$ then
$\sin\theta\cos\tau-\sin\gamma\sin\tau=0$, which implies that one of
$\cos\gamma\sin\tau$, $\cos\theta\cos\tau$, or
$\sin\gamma\cos\tau+\sin\theta\sin\tau$ is non-zero. It follows that $(0,0)$
is a regular value for ${\bf G}_{0}$ and $V_{0}$ is a smooth surface and a
closed subset of $T\times S^{1}\times[-\frac{1}{2},\frac{1}{2}]$ which misses
its boundary and has trivial normal bundle. Thus $V_{0}$ is closed, compact,
and orientable. It follows from straightforward differential topology
arguments that after perhaps shrinking $s_{0}$, the composite

$\mathbb{V}\subset[-s_{0},s_{0}]\times T\times\mathbb{S}_{{{\bf
i}}}^{*}\to[-s_{0},s_{0}]$

is a trivial surface bundle.

The equation ${\bf G}_{0}(\gamma,\theta,\tau,\nu)=(0,0)$ is equivalent to

$\nu=0,~{}(\sin^{2}\gamma+\sin^{2}\theta)^{1/2}(\cos\tau,\sin\tau)=\pm(\sin\gamma,\sin\theta).$

For fixed $(\sin\gamma,\sin\theta)\neq(0,0)$, there are precisely two
solutions: $(\nu,\tau)=\pm(0,~{}\tau(\gamma,\theta))$, where

(3.14)
$(\cos(\tau(\gamma,\theta)),\sin(\tau(\gamma,\theta)))=\pm\frac{(\sin\gamma,\sin\theta)}{\sqrt{\sin^{2}\gamma+\sin^{2}\theta}}.$

On the other hand, if $(\sin\gamma,\sin\theta)=(0,0)$, there is precisely a
circle of solutions, namely

$\nu=0,~{}\tau\text{ arbitrary. }$

This proves the last statement in the corollary.

Hence the smooth, compact surface $V_{0}$ is the union of two open disjoint
four-punctured tori and four circles, the preimages of the fixed points in
$T$. It follows that $V_{0}$ has Euler characteristic negative eight. The
continuous path

$\gamma=\arcsin(t),\theta=0,\tau=0,\nu=0,~{}~{}t\in(-\tfrac{1}{2},\tfrac{1}{2})$

lies in $V_{0}$ and its endpoints lie in different sheets of the covering
space. Hence $V_{0}$ is connected, and therefore of genus five. It now follows
by the Riemann-Hurwitz formula that the quotient of $V_{0}$ by the
orientation-preserving free involution $\hat{\iota}$ has genus three. We
conclude that $\mathbb{V}/\hat{\iota}\to[-s_{0},s_{0}]$ is a smooth trivial
genus three surface bundle.

Exactly the same argument holds for
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{\prime},V_{s}^{\prime},\mathbb{V^{\prime}}$.
$\square$

##  4\. restriction to $P_{0}^{*}$

We continue our analysis of the restriction maps
$u_{s}=\pi_{0}\times\pi_{1}:V_{s}/\hat{\iota}=\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\to
P_{0}\times P_{1}$ (resp. $u^{\prime}_{s}$) in this section by proving that
for small non-zero $s$, the map
$\pi_{0}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\to
P_{0}$ misses the corners and has only embedded fold singularities in the
sense of Whitney [37]. In Section 4.1, we define appropriate terminology to
describe the singularities. In Section 4.2, we analyze the map to $P_{0}$. The
corresponding analysis for the projection to $P_{1}$ is carried out in Section
5.

###  4.1. Fold singularities and bifold Lagrangian immersions of a surface
into a product of surfaces.

We remind the reader of the definition of a fold singularity and introduce the
concept of bifold Lagrangian immersions of a surface into a product of two
symplectic surfaces. We begin by recalling aspects of Whitney’s theory about
smooth maps between surfaces [37].

If $f:F\to G$ is a smooth map between surfaces, we say $x\in F$ is a
singularity if $\ker df_{x}\neq 0$. The map $f:F\to G$ is said to have only
fold singularities provided the 1-jet of $f$ misses the rank zero 1-jets and
meets the submanifold of rank one 1-jets transversely, and the set of
singularities, which in this case is called the fold locus, is a smooth one-
dimensional submanifold of $F$ which is immersed by $f$. These conditions can
be restated as

  1. (1)

at each $x$ in the fold locus, $\operatorname{rank}(df_{x})=1$,

  2. (2)

$0$ is a regular value of $\det(df)$, and

  3. (3)

$f$ immerses the fold locus.

The terminology fold comes from Whitney’s theorem [37] which asserts that if
these three conditions hold at a singular point $x$, then there exist local
coordinates around $x\in F$ and $f(x)\in G$ such that $f$ takes the form

$(u,v)\mapsto(u^{2},v).$

The fold locus corresponds to the set where $u=0$ in this local form.

If $f$ furthermore self-transversely immerses the fold locus into $G$, then
$f$ is stable, that is, every smooth map $C^{\infty}$ close to $f$ is equal to
$h_{G}\circ f\circ h_{F}$ for some diffeomorphisms $h_{F}:F\to F,h_{G}:G\to
G$. These facts are stated and proven carefully in [15, Section III.4]. Note
that if $f$ has only fold singularities and $f$ embeds the fold locus, then
$f$ is stable; in this situation, we say $f$ has only embedded fold
singularities, and we call the image of the fold locus under $f$ the fold
image.

Let $F_{0},F_{1}$ and $F$ be smooth surfaces, and let $f=f_{0}\times
f_{1}:F\to F_{0}\times F_{1}$ be a smooth map. Then $f$ is called a bifold
immersion if $f$ is an immersion with the property that $f_{0}$ and $f_{1}$
have only fold singularities, and their fold loci coincide. For example, the
map $(u,v)\mapsto(u,v^{2},u+v,v^{2})$ is a bifold immersion (in fact,
embedding) ${\mathbb{R}}^{2}\to{\mathbb{R}}^{2}\times{\mathbb{R}}^{2}$, with
fold locus $\\{(u,0)\\}$ and embedded fold image $\\{(u,0,u,0)\\}.$

Not every immersion into a product of surfaces is a bifold immersion. For
example, $(u,v)\mapsto(u^{4},v,u^{4},u+v)$ is an embedding of
${\mathbb{R}}^{2}$ into ${\mathbb{R}}^{2}\times{\mathbb{R}}^{2}$ which is not
a bifold immersion because $0$ is not a regular value of $\det df_{0}$ or
$\det df_{1}$. Another interesting non-bifold is the immersion
$(u,v)\mapsto(u\cos v,u\sin v,\tfrac{u^{2}}{2},u+v)$ whose projection to the
first ${\mathbb{R}}^{2}$ is constant, and hence not an immersion, on the
singular set $u=0$, while its projection to the second ${\mathbb{R}}^{2}$ has
only embedded fold singularities, with fold locus $v=0$. We note that these
examples are Lagrangian immersions into
${\mathbb{R}}^{2}\times{\mathbb{R}}^{2}$ with respect to the symplectic form
$-dx_{1}\wedge dy_{1}+dx_{2}\wedge dy_{2}$.

While the examples above demonstrate that a Lagrangian immersion of a surface
into a product of symplectic surfaces need not be a bifold immersion, the
following lemma shows that the Lagrangian immersion property does put
constraints on the behaviors of the factor maps. Given symplectic surfaces
$(F_{0},\omega_{0})$ and $(F_{1},\omega_{1})$, equip $F_{0}\times F_{1}$ with
the symplectic form $-{\rm proj}_{0}^{*}(\omega_{0})+{\rm
proj}_{1}^{*}(\omega_{1}).$ We use the abbreviated notation $F_{0}^{-}\times
F_{1}$.

######  Lemma 4.1.

Suppose $f=(f_{0},f_{1}):F\to F_{0}^{-}\times F_{1}$ is a Lagrangian immersion
of a surface into a product of symplectic surfaces.

Then, for each $x\in F$, the derivatives $df_{0,x}$ and $df_{1,x}$ have equal
non-zero rank. For those $x\in F$ satisfying $\text{\rm rank }df_{0,x}=1$, the
kernels of $df_{1,x}$ and $df_{0,x}$ are transverse.

###### Proof.

Since $f$ is an immersion, the differential of $df_{x}$ is injective and maps
the tangent space $T_{x}F$ to a subspace of the direct sum of subspaces
$T_{f_{0}(x)}F_{0}\oplus T_{f_{1}(x)}F_{1}$. Hence:

$0=\ker df_{x}=\ker df_{0,x}\cap\ker df_{1,x}.$

If $df_{0,x}$ has rank zero or one, then the Lagrangian hypothesis implies
that

$0=f^{*}(-{\rm proj}_{0}^{*}(\omega_{0})+{\rm
proj}_{1}^{*}(\omega_{1}))_{x}=f_{1}^{*}(\omega_{F_{1}})_{x},$

and so $df_{1,x}$ cannot be an isomorphism, and similarly with the subscripts
$0,1$ reversed. Therefore, either $df_{0,x}$ and $df_{1,x}$ are both
isomorphisms, or both have rank one, with transverse kernels. $\square$

Although it is not used in the present article, we note that [38] proves that
each Lagrangian immersion of a surface into a product of surfaces is
Lagrangian regular homotopic to one whose compositions with the projection to
the factors are stable (in the sense of Whitney).

###  4.2. The critical set of
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s},\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{\prime}\xrightarrow{{\pi_{0}}}P_{0}$
for small non-zero $s$

The projection map $V_{0}\to T$ is complicated, as seen in Corollary 3.3; the
fibers over the four fixed points are circles, whereas over other points the
map is a trivial two-fold cover. We prove below that when $|s|\neq 0$ is
small, the map
$\widehat{\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}}_{s}=V_{s}\to
T$ has only embedded fold singularities and, furthermore, its image misses the
four fixed points. Composing with the quotient map $T\to P_{0}$ of the
involution $\iota$, one concludes that the map
$\pi_{0}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\to
P_{0}$ misses the four corner points and has only embedded fold singularities.
Since the arguments only depend on the first order Taylor expansion of the
first component of ${\bf G}_{s}$, the arguments work equally well for ${\bf
G}^{\prime}_{s}$ with no modification.

We produce coordinate charts on four open subsets of $T\times{\mathbb{S}_{{\bf
i}}}$ that make it easier to analyze the tubular neighborhoods of the four
circles in $V_{0}$ which are the preimages of the fixed points in $T$.

To motivate the change of coordinates, note that when $s=0$, all the points in
$V_{0}$ satisfy $\nu=0$. Consider the remaining conditions on
$(\gamma,\theta,\tau)$ in the three-torus. The four two-tori in the three-
torus defined by the equations $\gamma=\pm\tfrac{\pi}{2}$ and
$\theta=\pm\tfrac{\pi}{2}$ decompose the three-torus into four disjoint solid
tori. One is

$\\{(\gamma,\theta,\tau)\mid|\gamma|<\tfrac{\pi}{2},|\theta|<\tfrac{\pi}{2}\\},$

and the others are obtained from this one by shifting $\gamma$ or $\theta$ by
$\pi$. The formula $(\gamma,\theta,\tau)\mapsto(\sin\gamma,\sin\theta,\tau)$
defines a diffeomorphism from each of these solid tori to
$(-1,1)\times(-1,1)\times{\mathbb{R}}/2\pi{\mathbb{Z}}$. The surface $V_{0}$
intersects each of these solid torus in an annulus. From Equation (3.10),
under these diffeomorphisms, these four annuli correspond to twisted annuli

$\\{(u,v,\tau)\mid-u\sin\tau+v\cos\tau=0\\}=\\{(u,v,\tau)\mid(u,v)\perp(-\sin\tau,\cos\tau)\\}.$

Our new coordinate system on each of these solid tori will be of the form
$(x,y,\tau)$ where the twisted annulus is defined by the equation $y=0$.

It is convenient in the calculations below to index the four solid tori by
pairs of signs,

(4.1) $(\epsilon_{\gamma},\epsilon_{\theta})\in\\{\pm 1\\}^{2}\text{ such that
}\cos\gamma=\epsilon_{\gamma}\sqrt{1-\sin^{2}\gamma}\text{ and
}\cos\theta=\epsilon_{\theta}\sqrt{1-\sin^{2}\theta},$

on the solid torus.

Specifically, on each solid torus, define functions $x$ and $y$ of
$(\gamma,\theta,\tau)$ by

(4.2) $\begin{pmatrix}x\\\ y\end{pmatrix}=\begin{pmatrix}\cos\tau&\sin\tau\\\
-\sin\tau&\cos\tau\end{pmatrix}\begin{pmatrix}\sin\gamma\\\
\sin\theta\end{pmatrix}=R_{-\tau}\begin{pmatrix}\sin\gamma\\\
\sin\theta\end{pmatrix},$

where

$R_{t}=\begin{pmatrix}\cos t&-\sin t\\\ \sin t&\cos t\end{pmatrix}$

Then the map $(\gamma,\theta,\tau)\mapsto(x,y,\tau)$ is a diffeomorphism from
each of the open solid tori in $T\times{\mathbb{R}}/2\pi{\mathbb{Z}}$ to an
open solid torus in ${\mathbb{R}}^{2}\times{\mathbb{R}}/2\pi{\mathbb{Z}}$. (In
terms of the coordinates $(x,y,\tau,\nu)$, the zero set of Equation (3.10) is
given by $\nu=0,y=0$.) The inverse diffeomorphism expresses $\gamma$ and
$\theta$ in terms of $x,y,\tau$, namely:

(4.3) $\begin{pmatrix}\gamma\\\ \theta\end{pmatrix}=A\circ
R_{\tau}\begin{pmatrix}x\\\ y\end{pmatrix},$

with $A=\arcsin\times\arcsin$ interpreted appropriately depending on
$(\epsilon_{\gamma},\epsilon_{\theta})$.

Going forward, fix $(\epsilon_{\gamma},\epsilon_{\theta})\in\\{\pm 1\\}^{2}$.
On the part of $\mathcal{D}$ where $(\gamma,\theta,\tau)$ is in the
corresponding solid torus, Equation (3.7) implies that, in the new
coordinates,

${\bf G}(s,x,y,\tau,\nu)=(y,\nu)+sR$

for some analytic function $R(s,x,y,\tau,\nu)$. The analytic implicit function
theorem now implies the following lemma.

######  Lemma 4.2.

After shrinking $s_{0}$ if needed, for each
$(\epsilon_{\gamma},\epsilon_{\theta})\in\\{\pm 1\\}^{2}$, there exist an
$x_{0}>0$, and two analytic functions

$Y(s,x,\tau),N(s,x,\tau):[-s_{0},s_{0}]\times[-x_{0},x_{0}]\times{\mathbb{R}}/2\pi{\mathbb{Z}}\to{\mathbb{R}},$

so that, for all $|s|\leq s_{0}$ and $|x|\leq x_{0}$,

${\bf G}_{s}(x,y,\tau,\nu)=0\text{ if and only if
}(y,\nu)=(Y(s,x,\tau),N(s,x,\tau)).$

Moreover, there exists an analytic function $R_{6}(s,x,\tau)$ so that

$Y(s,x,\tau)+2s\cos\gamma(x,Y(s,x,\tau),\tau)\cos\theta(x,Y(s,x,\tau),\tau)=s^{2}R_{6}(s,x,\tau).$

$\square$

We denote by $k_{s}:V_{s}\to T$ the composition map in the following diagram

(4.4) ${V_{s}={\bf G}_{s}^{-1}(0,0)}$${T\times{\mathbb{S}_{{\bf
i}}}}$${T}$$\scriptstyle{k_{s}}$$\scriptstyle{\rm proj}$

When $s=x=0$, for the four $(\epsilon_{\gamma},\epsilon_{\theta})$ choices,
Lemma 4.2 describes a parameterization by
$\tau\in{\mathbb{R}}/2\pi{\mathbb{Z}}$ of the circle in $V_{0}$ projecting to
the fixed point in $T$ corresponding to
$(\epsilon_{\gamma},\epsilon_{\theta})$. Specifically, these four circles lie
in the critical set of $k_{0}$, but $k_{0}$ is a covering map away from these
circles, with each sheet of the covering a graph of a smooth function. Thus
the critical set of $k_{0}$ is precisely the union of these four circles.

We now identify the critical set of $k_{s}$, for $|s|$ small but nonzero, and
identify the image of this critical set in $T$. Recall that the smooth family
$V_{s}={\bf G}_{s}^{-1}(0,0)$ coincides with
$\widehat{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}_{s}$
when $s\neq 0$. This allows us to analyze the map
$\pi_{0}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\to
P_{0}$, due to the commutative diagram (2.15).

######  Proposition 4.3.

After shrinking $s_{0}$ if needed, then the image of the composition $k_{s}$
in Equation (4.4) misses the four elliptic fixed points for all $0<|s|\leq
s_{0}$. Moreover, the critical set of this map is a disjoint union of four
circles, transversely cut out by the 1-jet of the map $k_{s}$. The four
critical circles are embedded under $k_{s}$ with images that are simple closed
curves encircling the four elliptic fixed points. In other words, $k_{s}$ has
only embedded fold singularities and the four circles making up the fold image
enclose the four elliptic fixed points.

###### Proof.

Because $k_{0}$ is a local diffeomorphism except along the critical circles,
it is sufficient to work in neighborhoods of the critical circles. Consider
the pair $(\epsilon_{\gamma},\epsilon_{\theta})\in\\{\pm 1\\}^{2}$ associated
to a fixed point. Referring to Equation (4.3), since each of the four branches
$A$ is a local diffeomorphism near $(0,0)$, it suffices to prove that, for
small enough $s$,

$\Gamma_{s}:[-x_{0},x_{0}]\times
S^{1}\to{\mathbb{R}}^{2},~{}\Gamma_{s}(x,\tau)=R_{\tau}\begin{pmatrix}x\\\
Y(s,x,\tau)\end{pmatrix}=:\begin{pmatrix}\sin\gamma(s,x,\tau)\\\
\sin\theta(s,x,\tau)\end{pmatrix}$

has a circle fold locus which embeds in ${\mathbb{R}}^{2}$ enclosing the
origin.

The second assertion in Lemma 4.2 implies that

(4.5) $Y(0,x,\tau)=0\text{ for all }(x,\tau).$

Moreover,
$\cos(\gamma(0,0,\tau))\cos(\theta(0,0,\tau))=\epsilon_{\gamma}\epsilon_{\theta}=\pm
1$ for all $\tau$, so that $\tfrac{\partial Y}{\partial s}(0,x,\tau)\neq 0$
for all $(x,\tau)$. Hence for all small enough non-zero $s$, $Y(s,x,\tau)\neq
0$. This implies the first statement in Proposition 4.3.

We turn next to the examination of the critical set of $k_{s}$, restricted to
the solid torus indexed by $(\epsilon_{1},\epsilon_{2})$. Denote this critical
set by $C_{s}^{\epsilon_{\gamma},\epsilon_{\theta}}$. By definition,
$C_{s}^{\epsilon_{\gamma},\epsilon_{\theta}}$ is the zero set of

(4.6) $\det(d\Gamma_{s})=\det(\tfrac{\partial\Gamma_{s}}{\partial
x},\tfrac{\partial\Gamma_{s}}{\partial\tau})=\det\left(R_{\tau}\begin{pmatrix}1&-Y\\\
\tfrac{\partial Y}{\partial x}&\tfrac{\partial
Y}{\partial\tau}+x\end{pmatrix}\right)=\tfrac{\partial
Y}{\partial\tau}+x+Y\tfrac{\partial Y}{\partial x}.$

Equation (4.5) implies that
$\tfrac{\partial{\left(\det(d\Gamma_{s})\right)}}{\partial x}=1$ when $s=0$,
so $0$ is a regular value of $\det(d\Gamma_{s})$ for all small enough $s$.
Equivalently, the 1-jet of $\Gamma_{0}$ cuts out
$C_{0}^{\epsilon_{\gamma},\epsilon_{\theta}}$ transversely (see Section 4.1).
The implicit function theorem implies the existence of an analytic function
$\widehat{X}(s,\tau)$, defined for all $s$ small enough and all $\tau\in
S^{1}$, so that

$\det(d\Gamma_{s})(s,x,\tau)=0\text{ if and only if }x=\widehat{X}(s,\tau).$

Shrink $s_{0}$ if needed so that $0$ is a regular value of $\det(d\Gamma_{s})$
and $\widehat{X}(s,\tau)$ is defined for all $|s|\leq s_{0}$.

Set

$\widehat{Y}(s,\tau)=Y(s,\widehat{X}(s,\tau),\tau).$

Then, for fixed $s$ with $|s|\leq s_{0}$, the map

(4.7) $e^{\tau{{\bf
i}}}\mapsto(s,\widehat{X}(s,\tau),\widehat{Y}(s,\tau),\tau,N(s,\widehat{X}(s,\tau),\tau))\in
V_{s}$

parameterizes the critical circle
$C_{s}^{\epsilon_{\gamma},\epsilon_{\theta}}$. It follows that the restriction
of $\Gamma_{s}$ to the critical circle
$C_{s}^{\epsilon_{\gamma},\epsilon_{\theta}}$ is parameterized as a map
$\alpha_{s}:S^{1}\to T$ and expressed in the local coordinates
$(\sin\gamma,\sin\theta)$ near the fixed point as

(4.8) $\alpha_{s}(e^{\tau{{\bf
i}}})=R_{\tau}\begin{pmatrix}\widehat{X}(s,\tau)\\\
\widehat{Y}(s,\tau)\end{pmatrix}.$

We note here that, without any further analysis, the critical set when $s=0$
is given by $x=y=0$, parameterized by $\tau$, so

(4.9) $\widehat{X}(0,\tau)=0\text{ and
}\widehat{Y}(0,\tau)=Y(0,\widehat{X}(0,\tau),\tau)=0.$

We use the following lemma to finish the proof of Proposition 4.3.

######  Lemma 4.4.

Given any $(\epsilon_{\gamma},\epsilon_{\theta})\in\\{\pm 1\\}^{2}$ and
$|s|\leq s_{0}$, there exist analytic functions $T_{1}(s,\tau)$ and
$T_{2}(s,\tau)$ so that

$\widehat{X}=s^{2}T_{1}\text{ and
}\widehat{Y}=-2\epsilon_{\gamma}\epsilon_{\theta}s+s^{3}T_{2}.$

###### Proof.

From the formula in Equation (4.3) for $\gamma$ and $\theta$ in terms of
$x,y,\tau$,

$\gamma(x,Y(x,s,\tau),\tau)=\ell_{1}\pi+O(x,s)\mbox{ and
}\theta(x,Y(x,s,\tau),\tau)=\ell_{2}\pi+O(x,s)\mbox{ for some
}\ell_{i}\in{\mathbb{Z}}.$

Therefore,

$\cos\left(\gamma(x,Y(x,s,\tau),\tau)\right)\cos\left(\theta(x,Y(x,s,\tau),\tau)\right)=\epsilon_{\gamma}\epsilon_{\theta}+O(x^{2},sx,s^{2}).$

It follows from the implicit formula for $Y(s,x,\tau)$ in Lemma 4.2 that

(4.10)
$Y(x,s,\tau)=-2s\epsilon_{\gamma}\epsilon_{\theta}+O(sx^{2},s^{2}x,s^{3}),$

which implies that both $\tfrac{\partial Y}{\partial x}(s,x,\tau)$ and
$\tfrac{\partial Y}{\partial\tau}(s,x,\tau)$ are $O(sx,s^{2})$.

Equation (4.9) implies that $\widehat{X}(s,\tau)=O(s)$, from which we conclude
that

$\tfrac{\partial Y}{\partial x}(s,\widehat{X}(s,\tau),\tau)=O(s^{2})\mbox{ and
}\tfrac{\partial Y}{\partial\tau}(s,\widehat{X}(s,\tau),\tau)=O(s^{2}).$

Using the defining equation for $\widehat{X}$ (i.e., Equation (4.6) equal to
zero), it follows that

$\widehat{X}(s,\tau)=-\tfrac{\partial
Y}{\partial\tau}(s,\widehat{X}(s,\tau),\tau)-Y(s,\widehat{X}(s,\tau),\tau)\tfrac{\partial
Y}{\partial x}(s,\widehat{X}(s,\tau),\tau)=O(s^{2}).$

Also, Equation (4.10) now implies

$\widehat{Y}(s,\tau)=Y(s,\widehat{X}(s,\tau),\tau)=-2s\epsilon_{\gamma}\epsilon_{\theta}+O(s^{3}).$

$\square$

With Lemma 4.4 in place, Equation (4.8) can be written in the form

(4.11)
$\alpha_{s}(\tau)=2s\epsilon_{\gamma}\epsilon_{\theta}\begin{pmatrix}-\sin\tau\\\
\cos\tau\end{pmatrix}+s^{2}R_{\tau}\begin{pmatrix}T_{1}\\\
sT_{2}\end{pmatrix}.$

By further shrinking $s_{0}>0$ if necessary, $0<|s|\leq s_{0}$ implies

  1. (1)

$\alpha_{s}$ misses the fixed point, and

  2. (2)

the smooth map

$\tfrac{\alpha_{s}(\tau)}{\|\alpha_{s}(\tau)\|}=\tfrac{\alpha_{s}(\tau)/s}{\|\alpha_{s}(\tau)/s\|}:S^{1}\to
S^{1}$

is defined and is a diffeomorphism.

Therefore, under the mapping $k_{s}:V_{s}\to T$, the critical circle
$C_{s}^{\epsilon_{\gamma},\epsilon_{\theta}}$ encircles the fixed point
corresponding to $(\epsilon_{\gamma},\epsilon_{\theta})$ via Equation (4.1).
This concludes the proof of Proposition 4.3. $\square$

In the following corollary, a ray emerging from a corner refers to a path in
$P_{0}$ of the form $[\gamma(t),\theta(t)]=[\ell_{1}\pi+tx,\ell_{2}\pi+ty]$,
$t\in[0,\tfrac{\pi}{2})$, $\ell_{i}\in{\mathbb{Z}},(x,y)\in S^{1}$.

######  Corollary 4.5.

After shrinking $s_{0}$ if needed, for all $s$ with $0<|s|\leq s_{0}$, the map

$\pi_{0}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\to
P_{0}$

misses the corners and has only embedded fold singularities. The fold locus
consists of four critical circles. Each corner is enclosed by its
corresponding the fold image circle. Any ray in $P_{0}$ emerging from a corner
intersects the image circle encircling that corner transversely. The
restriction of $\pi_{0}$ to the complement of the fold locus is a trivial two-
fold cover of its image in $P_{0}^{*}$.

###### Proof.

The fact that $\pi_{0}$ misses the corners follows from the fact that $k_{s}$
misses the fixed points. Given that the quotient
$T^{*}=T\setminus\\{\mbox{fixed points}\\}\to P_{0}^{*}=T^{*}/\hat{\iota}$ is
a local diffeomorphism, it follows from the diagram (2.15) that $\pi_{0}$ has
only fold singularities.

The diagram:

(4.12)
${V_{s}}$${V_{s}}$${T}$${T}$$\scriptstyle{\hat{\iota}}$$\scriptstyle{k_{s}}$$\scriptstyle{k_{s}}$$\scriptstyle{\iota}$

commutes for all $s$, and hence, when $s\neq 0$, the critical set of $k_{s}$
is invariant under $\hat{\iota}$. Since the fold image
$k_{s}(C_{s}^{\epsilon_{\gamma},\epsilon_{\theta}})$ converges to the corner
corresponding to $(\epsilon_{\gamma},\epsilon_{\theta})$,
$C_{s}^{\epsilon_{\gamma},\epsilon_{\theta}}$ must be invariant under the
involution $\hat{\iota}$. Hence each of the four quotient circles
$C_{s}^{\epsilon_{\gamma},\epsilon_{\theta}}/\hat{\iota}$ maps
diffeomorphically to $P_{0}^{*}$, encircling its corresponding corner.

Because of the local structure of a fold map, for any non-zero $s$ there
exists an open neighborhood $W$ of the the fold locus such that the
restriction of $\pi_{0}$ to $W$ is two-to-one off the fold locus. When $s=0$,
$\pi_{0}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{0}\to
P_{0}$ is two-to-one away from the critical set. The last claim about
$\pi_{0}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\to
P_{0}$ follows. $\square$

######  Corollary 4.6.

The statement in Corollary 4.5 holds as well when
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}$
is replaced by
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{\prime}$.

###### Proof.

To begin with, the proof of Lemma 4.2 depends only on the first order
expansion of ${\bf G}_{s}$, so it works for ${\bf G}^{\prime}_{s}$, too. It
follows that the statement in Proposition 4.3 holds with $V_{s}={\bf
G}_{s}^{-1}(0,0)$ replaced by $V_{s}^{\prime}=({\bf
G}^{\prime}_{s})^{-1}(0,0).$ $\square$

##  5\. Restriction to $P_{0}\times P_{1}$

Having described the restriction
$\pi_{0}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\to
P_{0}$, we now describe the restriction
$\pi_{1}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\to
P_{1}$ to the second pillowcase, making use of an involution of the tangle,
$U$, which exchanges its boundary components. The strategy used is to express
$\pi_{1}$ as a composite $\pi_{1}=(\Psi\circ\Theta)\circ\pi_{0}\circ U_{s}$
where
$U_{s}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\to\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}$
is the involution induced by $U$,
$\pi_{0}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\to
P_{0}$ continues to denote the composition of projection onto $T$ with the
quotient $T\to T/\iota$, and $\Theta:P_{0}^{*}\to P_{0}^{*}$,
$\Psi:P_{0}^{*}\to P_{1}^{*}$ are $s$-independent explicit symplectomorphisms.
In addition, we introduce a pair of involutions $W_{1},W_{2}$ on
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}$
and
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{\prime}$,
used to streamline the proof of the main result.

###  5.1. The involution $U$

Denote by

$U:S^{2}\times[0,1]\to S^{2}\times[0,1]$

the involution

(5.1) $U(x,y,z,t)=(-x,y,z,1-t).$

The involution $U$ is orientation-preserving, has fixed point set the circle
$\\{x=0\\}\times\\{\tfrac{1}{2}\\}$, and interchanges the two boundary
components. Moreover, the two-sphere $S^{2}\times\\{\tfrac{1}{2}\\}$ is
$U$-invariant.

Figure 4. The earring and bypass tangles with their perturbation curves, loops
generating $\Pi$ and $\Pi^{\prime}$, and the involution $U$.

The earring and bypass tangles, endowed with their perturbation curves, may be
isotoped to be $U$-invariant and to lie in $D^{2}\times I$, as illustrated in
Figure 4.

Figure 4 is simply a different perspective of Figure 3, and the illustrated
labelled loops based at a fixed point of $U$ correspond to the labelled based
loops in Figure 3. In particular, generators and relations in the
presentations given in Propositions 2.1 and 2.7 agree.

Rotating Figure 4 along the horizontal axis by the angle $\pi$ and keeping
track of where homotopy classes of based loops are sent establishes the
following lemma, which identifies the induced automorphism $U_{\\#}$ of $\Pi$
(resp. $\Pi^{\prime}$).

######  Lemma 5.1.

In terms of the presentations given in Propositions 2.1 and 2.7, for both
tangles,

$U_{\\#}(a)=\bar{d}=\bar{f}\bar{q}f\bar{a}\bar{b}\bar{p}bpq,~{}U_{\\#}(b)=\bar{d}\bar{c}d=(\bar{f}\bar{q}f\bar{a}\bar{b}\bar{p})\bar{b}(pba\bar{f}qf),~{}U_{\\#}(f)=\bar{f},$

Since $U$ preserves the two tori which bound tubular neighborhoods of the
perturbation curves, there exist loops $\gamma_{i}$ so that

$U_{\\#}(p)=\gamma_{1}\bar{p}\gamma_{1}^{-1},~{}U_{\\#}(\lambda_{p})=\gamma_{1}\lambda_{p}^{-1}\gamma_{1}^{-1},~{}U_{\\#}(q)=\gamma_{2}\bar{q}\gamma_{2}^{-1},~{}U_{\\#}(\lambda_{q})=\gamma_{2}\lambda_{q}^{-1}\gamma_{2}^{-1}$

In addition,

  * •

for the earring tangle, $U_{\\#}(w)$ is conjugate to $w$ and
$U_{\\#}(h)=\bar{h}\bar{w}$, and

  * •

for the bypass tangle $U_{\\#}(h)=\bar{a}h\bar{q}\bar{p}\bar{h}pq\bar{h}a$.
$\square$

Lemma 5.1 implies that the action of $U$ on $\operatorname{Hom}(\Pi,SU(2))$
(resp. on $\operatorname{Hom}(\Pi^{\prime},SU(2))$) preserves the
$s$-perturbed flat traceless representations, and, in the case of the earring
tangle, the $w_{2}$ condition, and thereby determines an involution which we
denote by
$U_{s}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\to\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}$
(and similarly
$U_{s}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}^{\prime}_{s}\to\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{\prime}$).
Explicitly, if $x\in\Pi$ and
$[\rho]\in\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}$

$(U_{s}(\rho))(x)=\rho(U_{\\#}(x)).$

Denote by

(5.2) $\widehat{U}:P_{0}\to P_{1}$

the isomorphism on character varieties induced by the restriction
$U|_{S^{2}\times\\{1\\}}:S^{2}\times\\{1\\}\to S^{2}\times\\{0\\}$.
Functoriality immediately implies the following (recall that we write
$u_{s}=\pi_{0}\times\pi_{1}$; see the comment following Lemma 3.1).

######  Lemma 5.2.

The diffeomorphism $U$ induces an analytic involution
$U_{s}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\to\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}$
for any $0<|s|<s_{0}$. The diagram

(5.3)
${\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}}$${\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}}$${P_{0}}$${P_{1}}$$\scriptstyle{U_{s}}$$\scriptstyle{\pi_{0}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\widehat{U}}$

commutes. The identical statement holds for
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}^{\prime}_{s}$.
$\square$

In the following,
$C_{s}\subset\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}$
denotes the fold locus,

(5.4)
$C_{s}=\\{c\in\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}~{}|~{}\ker(d\pi_{0})_{c}\neq
0\\}.$

Hence $C_{s}$ is the union of the four quotient circles
$C_{s}^{\epsilon_{\gamma},\epsilon_{\theta}}/\hat{\iota}$. The same holds for
the fold locus
$C_{s}^{\prime}\subset\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{\prime}$.

###  5.2. Relationship of $U_{s}$ to $\pi_{1}$

We next express
$\pi_{1}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\to
P_{1}^{*}$ in terms of $U_{s}$, $\pi_{0}$, and a fixed identification
$\Psi:P_{0}\cong P_{1}$.

Denote by $\psi:\Pi_{1}\to\Pi_{0}$ the isomorphism of peripheral subgroups
taking the ordered triple of free generators $d,c,f$ to the free generators
$a,b,f$. Note that $\psi$ is induced by the obvious homeomorphism from
$(S^{2},4)\times\\{1\\}$ to $(S^{2},4)\times\\{0\\}$, which is orientation-
reversing with respect to the boundary orientation on these two components.
Then set

(5.5) $\Psi:P_{0}\to P_{1},~{}\Psi=\psi^{*}$

the induced map on traceless character varieties.

Also define

(5.6) $\Theta:P_{0}\to P_{0},~{}\Theta([\gamma,\theta])=[\gamma,-\theta].$

Notice that $\Theta$ is orientation-reversing on $P_{0}^{*}$. In terms the
coordinates $P_{0}^{*}\subset{\mathbb{R}}^{3}$ given by the characters of the
loops $a,b,f$ (see Equation (2.5):

(5.7) $\Theta(x,y,z)=(x,y,2xy-z).$

######  Lemma 5.3.

  1. (1)

The map $\widehat{U}:P_{0}\to P_{1}$ factors as

$\widehat{U}=\Psi\circ\Theta.$

  2. (2)

The maps
$\pi_{1}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s},\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}^{\prime}_{s}\to
P_{1}^{*}$ factor as

$\pi_{1}=\Psi\circ\Theta\circ\pi_{0}\circ U_{s}.$

  3. (3)

The map $u_{s}$ factors as

$u_{s}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\xrightarrow{\pi_{0}\times(\Theta\circ\pi_{0}\circ
U_{s})}P_{0}^{*}\times P_{0}^{*}\xrightarrow{{\rm
Id}\times\Psi}P_{0}^{*}\times P_{1}^{*}$

and similarly for
$u^{\prime}_{s}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}^{\prime}_{s}\to
P_{0}^{*}\times P_{1}^{*}$.

###### Proof.

We show that $\Psi^{-1}\circ\widehat{U}=\Theta$, which implies the first
statement. Let $\widehat{U}_{\\#}:\Pi_{1}\to\Pi_{0}$ denote the isomorphism
induced on peripheral subgroups by the involution $U$, so that
$(\widehat{U}_{\\#})^{*}=\widehat{U}$. Using Lemma 5.1, we compute the action
of $\widehat{U}_{\\#}\circ\psi^{-1}$ on the peripheral subgroup $\Pi_{0}$:

$\widehat{U}_{\\#}\circ\psi^{-1}(a)=\widehat{U}_{\\#}(d)=\widehat{U}_{\\#}(\widehat{U}_{\\#}^{-1}(\bar{a}))=\bar{a}$
$\widehat{U}_{\\#}\circ\psi^{-1}(b)=\widehat{U}_{\\#}(c)=\widehat{U}_{\\#}(\widehat{U}_{\\#}^{-1}(\bar{a}\bar{b}a))=\bar{a}\bar{b}a,~{}\text{
and }$ $\widehat{U}_{\\#}\circ\psi^{-1}(f)=\widehat{U}_{\\#}(f)=\bar{f}.$

Hence

$(\widehat{U}_{\\#}\circ\psi^{-1})^{*}=\Psi^{-1}\circ\widehat{U}:P_{0}\to
P_{0}$

takes $[\gamma,\theta]\in P_{0}$ to the conjugacy class of

$a\mapsto\bar{{\bf i}},~{}b\mapsto\bar{{\bf i}}e^{\gamma{{\bf k}}}\bar{{\bf
i}}{{\bf i}}=e^{-\gamma{{\bf k}}}\bar{{\bf i}},~{}f\mapsto e^{\theta{{\bf
k}}}\bar{{\bf i}}.$

Conjugating by ${{\bf j}}$ yields $[\gamma,\theta]\mapsto[\gamma,-\theta]$,
which equals $\Theta$. This proves the first statement. The second and third
statements follow from the first and Lemma 5.2.

$\square$

Since $\Psi$ is induced by an orientation-reversing diffeomorphism between the
two four-punctured two-sphere boundary components, $\Psi$ restricts to a
symplectomorphism $\Psi:P_{0}^{*-}\to P_{1}^{*}$. For all $0<|s|<s_{0}$, the
first map in the third statement of Lemma 5.3,
$\pi_{0}\times(\Theta\circ\pi_{0}\circ
U_{s}):\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\to
P_{0}^{*-}\times P_{0}^{*-}$, is a Lagrangian immersion (see Theorem 5.5). The
second map, ${\rm Id}\times\Psi:P_{0}^{*-}\times P_{0}^{*-}\to
P_{0}^{*-}\times P_{1}^{*}$ is an ($s$-independent) symplectomorphism induced
by the natural identification of the two boundary components of $S^{2}\times
I$.

######  Proposition 5.4.

For $0<|s|\leq s_{0}$, the induced morphism
$U_{s}:\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}\to\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}$
is an orientation-reversing involution of a closed genus three surface. Each
of the four circles in the fold locus $C_{s}$ is $U_{s}$-invariant. The fold
locus separates
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}$
into two surfaces with boundary,

$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}=\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{+}\cup_{C_{s}}\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}^{-}_{s},$

each homeomorphic to a two-sphere with four open discs removed. Each surface
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{\pm}$
maps homeomorphically into $P_{0}^{*}$ by $\pi_{0}$ with image the complement
of a neighborhood of the corners, with boundary the fold image. Moreover,
$U_{s}(\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{+})=\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{+}$
and
$U_{s}(\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{-})=\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}_{s}^{-}$.
The identical statement holds for
$\operatorname{{{\mathchoice{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\displaystyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\textstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptstyle\natural$}}}{\raisebox{1.0pt}{\scalebox{2.3}[1.0]{$\scriptscriptstyle\natural$}}}}}}^{\prime}_{s}$.

Remark.  For concreteness, define