# Three dimensional higher-order raypath separation in a shallow-water
waveguide

Jiang Longyu<EMAIL_ADDRESS>Zhang Zhe<EMAIL_ADDRESS>The Laboratory of
Image Science and Technology, Southeast University, Nanjing 210096, China,
Centre de Recherche en Information Biomedicale Sino-Francais (LIA CRIBs),
Rennes, France.  Roux Philippe<EMAIL_ADDRESS>Institut
des Sciences de la Terre, Université Joseph Fourier, Centre National de la
Recherche Scientifique,1381 Rue de la Piscine, Saint-Martin d’Hères, France.

###### Abstract

Separating raypaths in a multipath shallow-water environment is a challenge
problem due to the interferences between them and colored noise existing in
ocean environment, especially for two raypaths arrive close to each other.
Thus, in this paper, a three dimensional (3D) higher-order raypath separation
in an array to array configuration is proposed. Performance tests using
simulation data in a multipath environment, real data obtained in an
ultrasonic waveguide and ocean shallow-water data, respectively, illustrate
that the proposed algorithm achieves a higher resolution and a stronger
robustness comparing to the existing algorithms.

##  I Introduction

Acoustic rays are multi-path propagation in shallow-water due to the
reflection and (or) the refraction by the surface of the ocean and the
continental shelf. The multiple raypaths cover more parts of the ocean and
provide more information than a single emitted signal. However, they also
produce interferences among them. Thus, separating the raypaths is necessary
in many important applications, such as passive sonar, ocean acoustic
tomography, etc. As a classical separation algorithm, the Multiple Signal
Classification (MUSIC) algorithm schimidt1986  was proposed to obtain a high-
resolution separation of the sources in a point to array configuration, which
is composed of a point source and a vertical receiver array. The MUSIC
algorithm improves separation resolution mainly exploiting the orthogonality
between the signal subspace and the noise subspace. However, it fails when the
signals are fully correlated or coherent. Jiang et el. jiang2017raypath
proposed a smoothing active wideband MUSIC (Smoothing-MUSICAL) algorithm,
which is an extension of the MUSIC algorithm to the case of separating fully
correlated or coherent wideband signals by using the spatial-frequency
smoothing. In addition, comparing to the MUSIC algorithm, the smoothing-
MUSICAL algorithm improves the separation resolution through adding the
arrival time of each raypath to the signal model as a new discrimination
parameter. Recently, a raypath separation algorithm based on the fourth order
cumulant jiang2017  is proposed in a point to array configuration. Higher
resolution and more robust separation is obtained due to the properties of
aperture extension and noise suppression of fourth-order cumulants while at a
cost of computation time. Sequentially, Jiang et el. propose a fast algorithm
using low-rank matrix approximation Jiang2018Fast  to reduce the computation
cost of the 4-smoothing-MUSICAL algorithm. On the other hand, Roux et el
Roux2008The  develop an array to array configuration, which is composed of a
source vertical array and a receiver vertical array. A double-beamforming
algorithm is introduced experimentally in the configuration and a high-
resolution tomography inversion Iturbe2008Ocean  is finally obtained through
using its separation results. Moreover, Touzé et el Touze2012Double  present
the double-capon algorithm and the double-MUSICAL algorithm under the
assumption of white Gaussian noise. To further improve both the resolution and
the robustness to the colored noise, we present a 3D higher-order raypath
separation algorithm in an array to array configuration in this paper. Its
performance improvement is achieved due to inheriting the merits of using both
the higher-order cumulants and the array to array configuration.

The rest of the paper is organized as follows. In Section 2, we give an
elaborate description of the 3D higher-order raypath separation algorithm. In
Section 3, we test the performance of the proposed algorithm using simulation
data in a multi-path environment, real data obtained in an ultrasonic tank and
ocean data, respectively. We conclude the paper in Section 4.

##  II The 3D higher-order raypath separation algorithm

###  II.1 Signal model

Assume the raypaths propagate in a double vertical array configuration, which
is composed of an emission array ($N$ sources) and a receiver array ($M$
sensors). The $P$ raypaths at the frequency $\nu$ produced by the $n^{th}$
source and received on the $m^{th}$ receiver is noted as $x_{m,n,\nu}$ and
modeled as follows.

$\centering
x_{m,n,\nu}=s_{\nu}\sum_{p=1}^{P}a_{p}e^{\Phi_{p}}+b_{m,n,\nu}\@add@centering$
(1)

where $s_{\nu}$ is the source spectrum,
$\Phi_{p}=-j2\pi\nu(T_{p}+(m-m_{0})\tau_{p}^{e}+(n-n_{0})\tau_{p}^{r})$,
$\tau_{p}^{e}=dsin(\theta_{p}^{e})/c$ and
$\tau_{p}^{r}=dsin(\theta_{p}^{r})/c$. $T_{p}$ notes the arrival time of the
$p^{th}$ raypath. $m_{0}$ ($n_{0}$) refers to the reference source (receiver).
$\tau_{p}^{e}$ ($\tau_{p}^{r}$) is the time delay for the $p^{th}$ raypath
propagates between two adjacent sources (receivers). $d$ is the interval
between two adjacent sources (receivers). $\theta_{p}^{e}$ ($\theta_{p}^{r}$)
is the direction of emission (reception) of the $p^{th}$ raypath. $a_{p}$
notes the amplitude of the $p^{th}$ raypath. $b_{m,n,\nu}$ refers to the
additive noise.

###  II.2 Data model

Based on the signal model built in above section, each signal received in the
double-array configuration is a data cube for all $N$ sources, $M$ sensors and
$F$ frequencies of the wideband signals considered in the algorithm. The data
model is built through concatenating its elements into a long vector. First,
the elements corresponding to the source $n^{th}$ at the frequency $\nu$ on
all the $M$ elements of the receiver array is concatenated into the vector as
follows ( $+$ notes transpose.) :

$\centering{\bf{x}}_{n,\nu}=[x_{1,n,\nu}\ldots
x_{M,n,\nu}]^{+}\@add@centering$ (2)

The elements corresponding to both the $N$ source and the $M$ sensors are
further concatenated into the following vector.

$\centering{\bf{x}}_{\nu}=[{\bf{x}}_{1,\nu}\ldots{\bf{x}}_{n,\nu}]^{+}\@add@centering$
(3)

Finally, considering all the $F$ frequencies, the long vector ${\bf{X}}$ is
obtained as follows.

$\centering{\bf{X}}=[{\bf{x}}_{\nu_{1}}\ldots{\bf{x}}_{\nu_{F}}]^{+}\@add@centering$
(4)

Similarly, the steering vector
$\mathbf{d}(\mathbf{\theta}_{p}^{e}\mathbf{\theta}_{p}^{r},\mathbf{T}_{p})$
and the additive noise $\mathbf{b}$ can be also concatenated into a long
vector. Thus, the received signal in the frequency domain can be written in
the following matrix form.

$\centering\mathbf{X}=\sum_{p=1}^{P}a_{p}\mathbf{d}(\mathbf{\theta}_{p}^{e},\mathbf{\theta}_{p}^{r},\mathbf{T}_{p})+\mathbf{b}=\mathbf{D}(\mathbf{\theta}_{p}^{e},\mathbf{\theta}_{p}^{r},\mathbf{T}_{p})+\mathbf{b}\@add@centering$
(5)

where

$\mathbf{X}=[\mathbf{x}_{\nu_{1}},\mathbf{x}_{\nu_{2}},\cdots,\mathbf{x}_{\nu_{F}}]^{+}$,
$\mathbf{x}_{\nu}=[\mathbf{x}_{1,\nu},\mathbf{x}_{2,\nu},\cdots,\mathbf{x}_{N,\nu}]^{+}\
\ (\nu=\nu_{1},\nu_{2},\cdots,\nu_{F})$, and
$\mathbf{x}_{n,\nu}=[x_{1,n,\nu},x_{2,n,\nu},\cdots,x_{M,n,\nu}]^{+}\
(n=1,2,\cdots,N)$.
$\mathbf{\theta}_{p}^{e}=[\theta_{1}^{e},\theta_{2}^{e},\cdots,\theta_{P}^{e}]^{+}$,
$\mathbf{\theta}_{p}^{r}=[\theta_{1}^{r},\theta_{2}^{r},\cdots,\theta_{P}^{r}]^{+}$,
$\mathbf{T}_{p}=[T_{1},T_{2},\cdots,T_{P}]^{+}$.

$\mathbf{d}(\mathbf{\theta}_{p}^{e},\mathbf{\theta}_{p}^{r},\mathbf{T}_{p})=[\mathbf{d}_{\nu_{1}}(\mathbf{\theta}_{p}^{e},\mathbf{\theta}_{p}^{r},\mathbf{T}_{p}),\mathbf{d}_{\nu_{2}}(\mathbf{\theta}_{p}^{e},\mathbf{\theta}_{p}^{r},\mathbf{T}_{p}),\cdots,\mathbf{d}_{\nu_{F}}(\mathbf{\theta}_{p}^{e},\mathbf{\theta}_{p}^{r},\mathbf{T}_{p})]^{+}$,
with
$\mathbf{d}_{\nu_{i}}(\mathbf{\theta}_{p}^{e},\mathbf{\theta}_{p}^{r},\mathbf{T}_{p})=[s_{\nu_{i}}e^{-j2\pi\nu_{i}(T_{p}+(1-n_{0})\tau(\theta_{p}^{e}))}\mathbf{d}(\mathbf{\theta}_{p}^{r})_{\nu_{i}},\cdots
s_{\nu_{i}}e^{-j2\pi\nu_{i}(T_{p}+(N-n_{0})\tau(\theta_{p}^{e}))}\mathbf{d}(\mathbf{\theta}_{p}^{r})_{\nu_{i}}]^{+}$
($i=1,2,\dots,F$) and
$\mathbf{d}(\mathbf{\theta}_{p}^{r})_{\nu_{i}}=[e^{-j2\pi\nu_{i}(1-m_{0})\tau(\theta_{p}^{r})},\cdots,e^{-j2\pi\nu_{i}(M-m_{0})\tau(\theta_{p}^{r})}]^{+}$.

$\mathbf{b}=[\mathbf{b}_{\nu_{1}},\mathbf{b}_{\nu_{2}},\cdots,\mathbf{b}_{\nu_{F}}]^{+}$,
with
$\mathbf{b}_{\nu}=[\mathbf{b}_{1,\nu},\mathbf{b}_{2,\nu},\cdots,\mathbf{b}_{N,\nu}]^{+}\
\ (\nu=\nu_{1},\nu_{2},\cdots,\nu_{F})$, and
$\mathbf{b}_{n,\nu}=[b_{1,n,\nu},b_{2,n,\nu},\cdots,b_{M,n,\nu}]^{+}\
(n=1,2,\cdots,N)$

###  II.3 Principle of the Algorithm

Using the data model built above, the trispectrum matrix of the received
signal can be theoretically computed according to the following equation.

$\begin{array}[]{rcl}\mathbf{C}=E\\{(\mathbf{X}\bigotimes\mathbf{X}^{*})(\mathbf{X}\bigotimes\mathbf{X}^{*})^{H}\\}\\\
-E\\{(\mathbf{X}\bigotimes\mathbf{X}^{*}){\\}}E\\{(\mathbf{X}\bigotimes\mathbf{X}^{*})^{H}\\}\\\
-E\\{(\mathbf{X}\mathbf{X}^{H})\\}\bigotimes
E\\{(\mathbf{X}\mathbf{X}^{H})^{*}\\},\end{array}$ (6)

However, in a real ocean experiment, it is costly to obtain several
realizations to compute the expectation of the stochastic process. Moreover,
the raypaths are reflected or refracted by the emitted signal thus they are
fully correlated or coherent, which will lead to a rank deficiency of the
trispectrum matrix. To resolve the problems, we use a three dimensional
smoothing technique Touze2012Double  to generate several realizations based on
the received signal. The smoothing is performed in the dimensions of the
emitted array, the receiver array and the frequency bins, respectively. That
is, the emitted (receiver) array is divided into $K_{e}$ ($K_{r}$) subarrays
for a length of $N_{e}^{s}=N-K_{e}+1$ ($N_{r}^{s}=M-K_{r}+1$). Similarly the
$K_{f}$ frequency subbands are respectively composed of $N_{f}^{s}=F-K_{f}+1$
frequency bins. Through these operation, we generate $K_{e}\times K_{r}\times
K_{f}$ small data cubes, whose dimensions are $N_{e}^{s}\times N_{r}^{s}\times
N_{f}^{s}$. The $\mathbf{X}_{i,j,k}$ is obtained through connecting the
elements of all the three dimensions into a long vector. To be easily
understood, the schematic diagram of the smoothing technique for the cumulants
is provided in Fig. 1.

Figure 1: Schematic diagram of the smoothing technique for the cumulants.

The expectation of the trispectrum matrix of the received signal is finally
estimated through an average of these trispectrum matrix computed using the
small data cubes, which is specifically described in Equation 4.

$\begin{array}[]{rcl}\widehat{\mathbf{C}}=\frac{1}{{{K}_{e}}{{K}_{r}}{{K}_{f}}}\sum\limits_{k=1}^{{{K}_{f}}}{\sum\limits_{j=1}^{{{K}_{e}}}{\sum\limits_{i=1}^{{{K}_{r}}}{{{\mathbf{C}}_{i,j,k}}}}}=\frac{1}{{{K}_{e}}{{K}_{r}}{{K}_{f}}}\sum\limits_{k=1}^{{{K}_{f}}}{\sum\limits_{j=1}^{{{K}_{e}}}}{\sum\limits_{i=1}^{{{K}_{r}}}}E\\{(\mathbf{X}_{i,j,k}\bigotimes\mathbf{X}_{i,j,k}^{*})(\mathbf{X}_{i,j,k}\bigotimes\mathbf{X}_{i,j,k}^{*})^{H}\\}\\\
-E\\{(\mathbf{X}_{i,j,k}\bigotimes\mathbf{X}_{i,j,k}^{*}){\\}}E\\{(\mathbf{X}_{i,j,k}\bigotimes\mathbf{X}_{i,j,k}^{*})^{H}\\}-E\\{(\mathbf{X}_{i,j,k}\mathbf{X}_{i,j,k}^{H})\\}\bigotimes
E\\{(\mathbf{X}_{i,j,k}\mathbf{X}_{i,j,k}^{H})^{*}\\},\end{array}$ (7)

Apply eigenvalue decomposition to $\widehat{\mathbf{C}}$

$\centering\widehat{\mathbf{C}}=\mathbf{U}\mathbf{\Lambda}{{\mathbf{U}}^{H}}=\sum\limits_{i=1}^{{{({{N}_{s}}{{M}_{s}}{{F}_{s}})}^{2}}}{{{\lambda}_{i}}{{\mathbf{u}}_{i}}\mathbf{u}_{i}^{H}=}\sum\limits_{i=1}^{{{P}^{2}}}{{{\lambda}_{i}}{{\mathbf{u}}_{i}}\mathbf{u}_{i}^{H}}+\sum\limits_{i={{P}^{2}}+1}^{{{({{N}_{s}}{{M}_{s}}{{F}_{s}})}^{2}}}{{{\lambda}_{i}}{{\mathbf{u}}_{i}}\mathbf{u}_{i}^{H}}\@add@centering$
(8)

where $\mathbf{U}$ is composed of ($N_{e}^{s}\times N_{r}^{s}\times
N_{f}^{s})^{2}$ eigenvectors and $\mathbf{\Lambda}$ is a diagonal matrix
containing the ($N_{e}^{s}\times N_{r}^{s}\times N_{f}^{s})^{2}$ eigenvalues.
$\mathbf{u}_{i}$ denotes the $i_{th}$ eigenvector. Arranging all the
eigenvalues from largest to smallest
$\lambda_{1}>\lambda_{2}>\cdots,>\lambda{{}_{(N_{e}^{s}\times N_{r}^{s}\times
N_{f}^{s})}}^{2}$, the $(N_{e}^{s}\times N_{r}^{s}\times N_{f}^{s})^{2}-P^{2}$
eigenvectors corresponding to the smallest eigenvalues project the noise
subspace ${\mathbf{U}}_{n}\mathbf{U}_{n}^{H}$ where

$\centering{\mathbf{U}}_{n}=\left[{\mathbf{u}}_{P^{2}+1},{\mathbf{u}}_{P^{2}+2},\cdots{{\mathbf{u}}_{(N_{e}^{s}\times
N_{r}^{s}\times N_{f}^{s})^{2}}}\right]\@add@centering$ (9)

Finally, the estimator for the cumulants in the double-array configuration is
constructed as follows.

$\centering{{P}_{Double\text{
}4-s-MUSICAL}}=\frac{1}{\mathbf{d}_{4}^{H}{{\mathbf{U}}_{n}}\mathbf{U}_{n}^{H}{{\mathbf{d}}_{4}}}\@add@centering$
(10)

with the steering vector
${{\mathbf{d}}_{4}}=\mathbf{d}(\theta_{p}^{r},\theta_{p}^{e},{{T}_{p}})\otimes{{\left(\mathbf{d}(\theta_{p}^{r},\theta_{p}^{e},{{T}_{p}})\right)}^{*}}$

##  III Performance test

In this section, the performance of the proposed algorithm is tested using
simulation data in a multi-path environment, small-scale data obtained in an
ultrasonic tank and ocean data Roux2008The , respectively. The test parameters
used in these experiments are shown in Table 1 and Table 2. $z_{s_{1}}\sim
z_{s_{M}}$ notes the depths in which the first source to the $M^{th}$ source
is fixed under the water while $z_{r_{1}}\sim z_{r_{N}}$ refers to the ranges
which the first receiver to the $N^{th}$ receiver cover. $D$ is the distance
between the reference source and the reference receiver. $c$ indicates the
acoustic velocity in which the raypaths propagate under the water. The central
frequency of the emitted signal is $\nu_{c}$ and its bandwidth is noted as
$\nu_{w}$. $\nu_{n}$ notes the number of frequencies used in each test
choosing the beginning frequency $\nu_{b}$ to the last frequency $\nu_{e}$.
$N_{s}$ refers to the number of samples used in each test in the time domain.

Figures 2 and 3 show the separation results of the proposed algorithm while
the contrast ones of the smoothing-MUSICAL algorithm and the 4-smoothing-
MUSICAL algorithm are also provided. Specifically, figure 2 (a), (c) and (e)
illustrate the separation results of a set of simulation data in the case that
five raypaths propagate between the emitter array and the receiver array,
which is composed of four receivers. The signal to noise ratio is equal to
2dB. The reference source (receiver) is located at 50m under the water. The
smoothing-MUSICAL algorithm provides a two-dimensional (2D) separation in a
plan of the arrival time and the direction of arrival (DOA) in figure 2 (a)
while the other two algorithms give a 3D separation with the additional
parameter: the direction of emission (DOE) in figure 2 (c) and figure 2 (e),
respectively. In figure 2 (a), (c) and (e), each spot corresponds to a raypath
and the black crosses mark the theoretical positions. It is obviously that the
smoothing-MUSICAL algorithm fails in finding the raypath with the DOA around
at $-5^{\circ}$ in figure 2 (a) and the raypath with the DOA around at
$0^{\circ}$ is deficient in the results of the double-MUSICAL algorithm in
figure 2 (c). In contrast, the proposed algorithm successfully separates all
the five raypaths without any artifact in figure 2 (e), where the deficient
raypaths in figure 2 (a) and figure 2 (c) are highlighted by the red circles.

Figure 2(b), (d) and (f) illustrate the performance contrast of the three
algorithms using a set of real data obtained at a small-scale ultrasonic tank.
The small-scale experiment reproduces the actual physical phenomena occurring
in the real ocean environment in a smaller scale inside the laboratory based
on the operation multiplying the frequency of the signals by a factor and
dividing the spatial distances by the same factor. Thus, the small-scale
experiment is a reduced cost and a totally controlled experiment.
Specifically, in this tank, a steel bar acts as the bottom, for which the
boundary conditions are nearly perfect at the water-bottom interface, and a
1.10m-long, 5.4-cm-deep acoustic waveguide is constructed Roux2014Inverting .
Two coplanar 64-element vertical line arrays (VLA) are placed and the central
frequency of the transducer is 1 or 3 MHz with a $50\%$ frequency bandwidth.
The transducer dimensions $0.75mm\times 12mm$ are used to make the linear
arrays omni-directional in the plane defined by the source-receiver arrays and
the beams collimated in the plane perpendicular to the waveguide axis. Figure
2(b), (d) and (f) show a group of separation results with a set of real data
obtained in the small-scale experiment. There are seven predictable raypaths
in the example, whose theoretical positions are indicated by the black
crosses. Figures 2(b) and 1(d) show the separation results using the
smoothing-MUSICAL algorithm and the double-MUSICAL algorithm, respectively. It
can be clearly seen that both of the two algorithms are deficient in detecting
one raypath with the DOA around at $5^{\circ}$. Comparing to them, the
proposed 3D higher order algorithm correctly find all the seven raypaths in
figure 1(f), where the red circle highlights the deficient raypath in figures
2(b) and 2(d).

Finally, the performance of the double-4-smoothing algorithm is tested using
ocean data. The experiment has been performed in July 2005 north of Elba
Island, Italy Roux2008The . It uses a similar experimental setup to the small-
scale ultrasonic experiment, although at a much larger scale. Two equally
spaced vertical linear arrays are implemented in 120 m water and the distance
between them is 4.071 km. The source array (SA) is composed of 29 transducers
covering 78 m and the receiver array (RA) has 32 hydrophones spanning 62m. The
central frequency of the transducers is 3.2 kHz with 1 kHz bandwidth. Figure 3
shows the contrast of separation results using a set of ocean data. There are
three expected rays, a surface-reflected ray and two refracted rays. Each
refracted ray has a turning point near the depth of maximum sound-speed
variability. Due to the sound speed variation described in the Ref.Roux2008The
, the three raypaths arrived at close time. Figure 3(a) shows that the
separation result of the smoothing-MUSICAL algorithm in the plan of the
arrival time and the DOA. It detects just a mixed spot for two of the three
raypaths due to their close arrival times and DOAs. Figure 3(b) shows that the
separation result of the double-MUSICAL algorithm in the plan of the emitted
angle and the DOA, which has a raypath deficiency with the DOA around at
$10^{\circ}$. However, the proposed algorithm gives an accurate separation for
the three raypaths shown in the plan of the emitted angle and the DOA in
Figure 3 (c) because of the different emitted angles and a strong robustness
of the proposed algorithm to the fluctuant ocean. Figure 3(d) displays the
three raypaths in Figure 3(a)-(c), which propagate between the centers of
source and receive arrays. Based on the performance contrasts with the
simulation data, small-scale data and ocean data, a conclusion is drawn that
the double-4-smoothing-MUSICAL algorithm enables to achieve an aperture
extension, a stronger robustness to the fluctuant environment and a resolution
improvement for the extension of cumulants-based algorithm to the double-array
configuration.

| $M$ | $N$ | $z_{s_{1}}\sim z_{s_{M}}(m)$ | $z_{r_{1}}\sim z_{r_{N}}(m)$ |  $d$ (m)
---|---|---|---|---|---
Simulation | 4 | 4 | $47.5\sim 52.5$ | $47.5\sim 52.5$ | 2.5
Small-scale experiment | 4 | 5 | $26.375\times 10^{-3}\sim 28.625\times 10^{-3}$ | $25.625\times 10^{-3}\sim 28.625\times 10^{-3}$ | $0.75\times 10^{-3}$
Ocean data | 4 | 4 | $93.098\sim 101.456$ | 94 $\sim$ 100 | 2
Table 1: The configuration parameters of simulation, small-scale experiment and the at-sea experiment. |  $\nu_{c}$ (Hz) |  $\nu_{w}$ (Hz) |  $\nu_{n}(\nu_{b}\sim\nu_{e})$ (Hz) | $N_{s}$ | $D(m)$ | $c\ (m/s)$
---|---|---|---|---|---|---
Simulation | $1.5\times 10^{3}$ | $5\times 10^{3}$ | 75 (0 $\sim$ $5\times 10^{3}$) | 135 | $2000$ | $1500$
Small-scale experiment | 1.2 $\times 10^{6}$ | 5 $\times 10^{6}$ | 150(0 $\sim$ 5 $\times 10^{6}$) | 132 | $1$ | $1473$
Ocean data | $3.2\times 10^{3}$ | $6\times 10^{3}$ | 50 (0 $\sim$ $6\times 10^{3}$) | 60 | $4.701\times 10^{3}$ | $1509$
Table 2: The test parameters used in the simulations, the small-scale
experiment and the at-sea experiment.

Figure 2: Separation results comparison of different algorithms using
simulation datas ((a), (c) and (e)) or using real data obtained in an
ultrasonic tank ((b), (d) and (f)). The black crosses denote the theoretical
values. The red circles highlight the raypaths used to differentiate the
separation ability of the algorithms. (a) and (b) show the results of the
smoothing-MUSICAL algorithm; (c) and (d) show the results of the double-
MUSICAL algorithm; (e) and (f) show the results of the double-4-MUSICAL
algorithm

Figure 3: Separation results comparison of different algorithms using ocean
datas. The black crosses denote the theoretical values. The red circles
highlight the raypaths used to differentiate the separation ability of the
algorithms (a) the smoothing-MUSICAL algorithm; (b) the double-MUSICAL
algorithm; (c) the double-4-MUSICAL algorithm; (d) Raypaths between the source
and the reference sensor for the three rays.

##  IV Conclusion

In this paper, we extended a higher-order algorithm to a double-array
configuration. Higher-resolution and more robust separation for close arrivals
even in a fluctuant ocean environment is achieved by the 3D algorithm. In
future work, reducing its computation complexity will be considered.

###### Acknowledgements.

This resarch has been supported by the National Natural Science Foundation of
China (Nos. 61871124 and 61876037), the State Key Laboratory of Acoustics,
Chinese Academy of Sciences (No. SKLA201604), and the Scientific Research
Foundation for the Returned Overseas Chinese Scholars. The Focused Acoustic
Forecasting experiment (FAF05) experiment was performed in collaborative
experiments with the NATO Underwater Research Centre (NURC), La Spezia, Italy,
with Mark Stevenson as Chief Scientist. Scientists who contributed to these
experiments include Tuncay Akal, W. A. Kuperman, W. H. Hodgkiss, H.C. Song,
B.D. Cornuelle, Piero Boni, Piero Guerrini, other NURC staff, and the officers
and crew of the RV Alliance.

## References

  * (1) R. Schmidt, ” Multiple emitter location and signal parameter estimation,” IEEE Transactions on Antennas and Propagation, vol. 34, no. 3, pp. 276-280, 1986.
  * (2) L. Jiang, P. Roux, and J. I. Mars, ” Raypath separation with a high-resolution algorithm in a shallow-water waveguide,” IEEE Journal of Oceanic Engineering, vol. 43, no. 1, pp. 119-130, 2018.
  * (3) L. Jiang, Y. Hong, P. Roux, J. Wu, and H. Shu, ” Active wideband higher-order raypath separation in multiple environment,” The Journal of the Acoustical Society of America, vol. 141, no. 1, pp. EL38-EL44, 2017.
  * (4) L. Jiang, W. Song, Z. Zhang, C.Yang, S. Wang, and P. Roux, ”Fast raypath separation based on low-rank matrix approximation in a shallow-water waveguide,” The Journal of the Acoustical Society of America, vol. 143, no. 1, pp. EL271-EL278, 2017.
  * (5) I. Iturbe, P. Roux, B. Nicolas, and J. Mars, ”Ocean acoustic tomography using a double-beamforming algorithm” IEEE Journal of Oceanic Engineering, vol. 123, no. 5, pp. 3912-EL278, 2008.
  * (6) G. Touzé, B. Nicolas, J. Mars, P. Roux, B. Oudompheng ”Double-Capon and double-MUSICAL for arrival separation and observable estimation in an acoustic waveguide,” Eurasip Journal on Advances in Signal Processing, vol. 143, no. 1, pp. EL271-EL278, 2017.
  * (7) Roux P and Nicolas B, ”Inverting for a deterministic surface gravity wave using the sensitivity-kernel approach”, Journal of the Acoustic Society of America, vol.135, no. 4, pp. 1789-1799, 2014.
  * (8) Roux P., Cornuelle B. D., Kuperman W. A, WS Hodgkiss, ”The structure of raylike arrivals in a shallow-water waveguide”, The Journal of the Acoustical Society of America, vol.124, no. 6, pp. 3430-3439, 2008.


Note that $qr$ times this number is the one found in Equation (4.5). Hence,
the symplectic-tropical curve we construct limit to a orbi-curve in the class
$qr\mathcal{A}$. In particular, for the symplectic-tropical curve to be a
smoothing of that orbiline in the class $\mathcal{A}$, we must have $q=1$ and
$r=1$. Indeed, by looking at the intersection with the Lagrangian spheres, we
will see that the actual smoothing (in case $qr=1$) is given by a _deformed_
symplectic-tropical curve.

####  4.4.2 Deforming Symplectic-Tropical Curves

As explained in the previous subsection, we need to be able to have control of
the intersection number of our symplectic-tropical curve $\mathscr{C}$ with
the Lagrangian 2-spheres that appear naturally for a pair of nodes lying
inside the same cut. For that, let us prove Proposition 4.17, with the
following notation.

Let $\mathscr{C}:\Gamma\longrightarrow P_{X}$ be a symplectic-tropical curve
in a ATF of $X$, represented by the ATBD $P_{X}$ and $\mathcal{N}$ a
neighborhood of $\mathscr{C}(\Gamma)$, as before. Consider a class of $n$
nodes of the ATBD inside the same cut, so that at least one of the nodes is in
$\mathscr{C}(\Gamma)$ and let $\mathcal{M}$ be a neighborhood of the cut. Let
$S_{1},\dots,S_{n-1}$ be Lagrangian spheres projecting inside the cut to
consecutive segments between the nodes. Also name $S_{0}=S_{n}=\emptyset$. Let
$m$ be the sum of the multiplicities of the leaves arriving at these $n$
nodes, and choose $d<n$ nodes, where $m\equiv d\mod n$.

######  Proposition 4.17.

There is a symplectic curve projecting into $\mathcal{N}\cup\mathcal{M}$ with
the property that it its intersection with $S_{1}\cup\dots\cup S_{n-1}$ is
precisely the $d$ chosen nodes. $\Box$

The reminder of this subsubsection is devoted to the proof of Proposition
4.17. The idea is that, since $m-d=kn$, we can place $\sigma_{j}$ disks for
Proposition 4.9, for $2\leq j\leq k+1$ at all the $n$ nodes. Naming
$n_{1},\dots,n_{n}$ be the $n$ nodes in a cut, in order, and $S_{i}$ a
Lagrangian sphere projecting to the cut between the consecutive nodes $n_{i}$,
$n_{i+1}$, we see that the signed intersection of $S_{i}$ with the collection
of $kn$ disks is zero, since the $S_{i}$ sphere only intersects the $k$ disks
around the $n_{i}$ and $n_{i+1}$ nodes, with opposite signs. Thus, it is clear
that, at least smoothly, we can pairwise cancel these intersections.
Nevertheless, before that, we need to connect the disks.

Let us focus now on how to construct in Lemma 8 a symplectic pair of pants
$\mathcal{P}$ that we can glue to the boundary of a pair of $\sigma_{j}$
symplectic 2-disks near the $n_{i}$ and $n_{i+1}$ nodes.

Figure 17: Amoebas of the surfaces $\Sigma_{j}^{i,i+1}$,
${\Sigma^{\prime}}_{j}^{i,i+1}$.

######  Lemma 8.

There is a symplectic disk $\Sigma_{j}^{i,i+1}$ in the cut neighborhood
$\mathcal{M}$, containing the two symplectic 2-disks $\sigma^{i}_{j}$ and
$\sigma^{i+1}_{j}$ disks as a subset, with boundary in a thickened 2-torus
$\mathbb{T}$, and whose class is two times the collapsing class of the nodes
via the identification $H_{1}(\mathbb{T};\Z)\cong H_{1}(T^{2};\Z)$.

Denote by $\mathcal{P}$ the pair of pants which is the closure of
$\Sigma_{j}^{i,i+1}\setminus(\sigma^{i}_{j}\cup\sigma^{i+1}_{j})$. $\Box$

Figure 17 shows on the left the amoeba corresponding to the
$\Sigma_{j}^{i,i+1}$ surface. The idea is to then isotope it to a different
surface ${\Sigma^{\prime}}_{j}^{i,i+1}$ with boundary still in $I\times
T^{2}$, such that the intersection with the Lagrangian 2-sphere $S_{i}$ is
empty.

Figure 18: (a) A dimer model for the symplectic pairs of pants $\mathcal{P}$.
(b) Dimer model for $\mathcal{P}^{\prime}$, illustrating the cycles in the
$S_{i}$ Lagrangian sphere, over the intesection between the boundary of the
amoeba of $\mathcal{P}^{\prime}$ and the amoeba of $S_{i}$ (a segment between
the nodes). (c) The dimer model for the part of the surface
${\Sigma^{\prime}}_{j}^{i,i+3}$ connecting the boundary of the $\sigma_{j}$
disk in the $n_{i+3}$ node with the boundary of
${\Sigma^{\prime}}_{j}^{i,i+2}$. (d) Coameaba of 2 non-intersecting
$\mathcal{P}^{\prime}$s. The suggested height is with respect to the $p_{2}$
coordinate.

######  Proof 4.18 (Proof of Lemma 8).

Let us assume the cut is vertical, and thus the collapsing cycles is given by
a $\theta_{2}$-constant curve in the $(\theta_{1},\theta_{2})$ coordinates of
$T^{2}$. Making a change of action-angles coordinates, if needed, we may
assume that the collapsing cycles corresponding to the $n_{i}$ nodes are
slightly phased-out. We thus draw the $T^{2}$ projection of the boundary of
the $\sigma_{j}$ 2-disks, recalling that it links once the horizontal cycle in
a thickened torus that they live in (recall construction before Proposition
4.9). We color these boundaries blue, and draw a dimer model that indicates
the coamoeba of the pair of pants $\mathcal{P}$ we will construct, as
illustrated in the first picture of Figure 18. We will color the other
boundary of $\mathcal{P}$ red. We number the components of the red curve in
the dimer model of Figure 18 (Left) as indicated, and sketch the profile of
its $p_{2}$ coordinate as indicated in Figure 19.

The $\partial\sigma_{j}$ curves can be taken sufficiently close to the
collapsing cycle $\partial\sigma_{1}$, so we assume its $\theta_{2}$ variation
is small enough with respect to the difference $p_{1}$ between the coordinate
of the red curve and the $p_{1}$ coordinate of the blue curve. (Essentially,
we take them small with respect to the size of $\mathcal{M}$.)

Figure 19: The profile of the $p_{2}$ coordinate as we move along
$\partial\Sigma_{j}^{i,i+1}$, $\partial{\Sigma^{\prime}}_{j}^{i,i+1}$. The
numbering corresponding to the ones on the leftmost picture of Figure 18.

The dimer model we are considering consists of two bi-gons and two tri-gons.
We carefully analyze the $(p_{1},p_{2})$-coordinates we associate to these
pieces, to ensure that get a symplectic pair of paints $\mathcal{P}$
connecting the blue boundaries to the red one. The corresponding amoebas,
together with the $\theta_{1}$\- and $\theta_{2}$-level sets, are indicated in
Figure 20 (Middle). There is a curve in the amoeba corresponding to the
vertices of the bi-gon, that will by crucial in further analysis. This curve
shall be named the pink curve, and in this case it is a horizontal segment.

Figure 20: Amoebas of the pieces of the pair of pants $\mathcal{P}$ and
$\mathcal{P}^{\prime}$ for the surfaces $\Sigma_{j}^{i,i+1}$,
${\Sigma^{\prime}}_{j}^{i,i+1}$.

Following analogous convention for the analysis of amoebas and coamoebas as in
the proof of Proposition 4.14, with $\rho_{i}$ corresponding to $p_{i}$ and
$\phi_{i}$ to $\theta_{i}$, we take the first curve $\xi^{1}$ corresponding to
a $\theta_{2}$-level set, oriented with $d\theta_{1}^{1}>0$ and the second
curve $\xi^{2}$ corresponding a $\theta_{1}$-level set, with the orientation
such that the $p_{1}$ coordinate is decreasing, i.e., $dp_{1}^{2}<0$.

First, let us study the bi-gon pieces. We assume that the $\theta_{2}$
variation is sufficiently small for the red curve as well, so if we fix the
norm of $dp^{2}$ as we travel, we then have $d\theta_{2}^{2}$ is small enough
away from the vertices. As indicated in Figure 20, we choose $\xi^{1}$ so that
$dp_{2}^{1}=0$ and $d\theta_{2}^{1}=0$. As we move along $\xi^{1}$ at the
bottom of the bi-gon, the $p_{1}$ coordinate increases up to the middle of the
horizontal curve at coamoeba and decreases from the middle until the end. The
logic is reversed as we move to the top of the bi-gon. So,
$\omega(\partial\xi^{1},\partial\xi^{2})=-d\theta_{1}^{1}dp_{1}^{2}>0$ (recall
equation (4.2)), as desired. The limit case where $\theta_{2}$ obtains the
maximum and the minimum in the bi-gon can be analyzed by replacing the
$\xi^{1}$ curve by the respective red and blue curves. Since it also
corresponds to maximum and minimum of the $p_{2}$ coordinate, we still have
$dp_{2}^{1}=0$ and $d\theta_{2}^{1}=0$, ensures positivity of
$\omega(\partial\xi^{1},\partial\xi^{2})$.

Second, we now turn our attention to the tri-gon, as depicted in Figure 20. We
first notice that the $p_{2}$ coordinates associated with the bottom vertex of
the tri-gon, must be different for the edge labeled $2$ and the edge labeled
$6$, the later being greater than the former, recall Figures 18 and 19. This
implies that we must not only consider one curve with that corresponding
$\theta_{2}$-coordinate constant, but rather a family of curves. The shaded
region in the bottom-middle picture of Figure 20 indicates the image of this
curves under the $(p_{1},p_{2})$-projection. The analysis at the top of this
part of the coamoeba is similar to before, with $dp_{2}^{1}=0$ and
$d\theta_{2}^{1}=0$ and
$\omega(\partial\xi^{1},\partial\xi^{2})=-d\theta_{1}^{1}dp_{1}^{2}>0$. (Note
that we do not remain stationary with $d\theta_{1}^{1}=0$ at the top part of
the coamoeba.) At the bottom part, we do have $dp_{2}^{1}>0$ as $\xi^{1}$
travels from the $2$ curve to the $6$ curve and we also have
$d\theta_{2}^{2}>0$, as we are traveling from the red curve to the blue curve
along $\xi^{2}$, with $dp_{1}^{2}<0$. That way, since $d\theta_{1}^{1}\geq 0$,
we have

$\omega(\partial\xi^{1},\partial\xi^{2})=-d\theta_{1}^{1}dp_{1}^{2}+d\theta_{2}^{2}dp_{2}^{1}>0.$

Gluing analogous models for the for the bottom part of the diagram in a
consistent way provides our symplectic pair of pants.

We now move towards the construction of the surface
${\Sigma^{\prime}}_{j}^{i,i+1}$ that does not intersect the Lagrangian sphere
$S_{i}$. Given our choice of coordinates, the Lagrangian sphere projects into
a vertical segment in the $(p_{1},p_{2})$ factor. In order to keep working
with the same coordinates in the region we want to do the modification of our
curve: we modify the ATBD, without changing our ATF, simply by reversing the
direction of the cuts associated with the nodes $n_{i+1},\dots,n_{n}$ (this
operation was named _transferring the cut_ in [54]). Note that we already made
this operation in Figures 17, 20. The rightmost diagrams of Figure 20
represents the projection of $S_{i}$ by a dashed segment, which is different
than the one we use to represent the cuts. Let us set $(p_{1},p_{2})=(0,0)$
the middle point of this dashed segment.

From Remark 4.8, we see that $\sigma^{i}_{j}$ and $\sigma^{i+1}_{j}$ both
intersect $S_{i}$ once, and with opposite signs. We consider disks
${\sigma^{\prime}}^{i}_{j}$ and ${\sigma^{\prime}}^{i+1}_{j}$, obtained from
$\sigma^{i}_{j}$ and $\sigma^{i+1}_{j}$ by carving out a neighborhood of the
intersection point with $S_{i}$, with their $(p_{1},p_{2})$ projection as
illustrated in Figure 17 (Right). We keep coloring their boundary blue, and
their coamoeba projection is similar to the ones we just analyzed, as it is
sufficiently close to the corresponding collapsing cycle with constant
$\theta_{2}$-coordinate. Thus, we can also build a dimer model as in Figure 18
(Left), to build a pair of paints $\mathcal{P}^{\prime}$, and we also color
the other boundary of $\mathcal{P}^{\prime}$ red.

Recall that we named a pink curve, and the $(p_{1},p_{2})$ image of the
vertices corresponding to the intersection of the blue and red curves in the
coamoeba coordinates $(\theta_{1},\theta_{2})$, which we will refer to as pink
vertices. It will play the following role in our construction. As before, we
focus on the top part of the coamoeba, the bottom part being symmetric under
the reflection around the $p_{2}=0$ coordinate (or at least having a symmetric
behaviour, since our coamoeba picture is not symmetric.) Then the pink curve
corresponding to the top bi-gon will be the graph of a non-increasing convex
function $p_{2}(p_{1})$, starting at a point with $p_{1}<0$, $p_{2}>0$,
becoming negative before $p_{1}$ becomes $0$, and eventually becoming constant
at some point where $p_{1}>0$. Hence the pink curve for the bottom bi-gon will
intersect this at some point with negative $p_{1}$ coordinate. We call $x$ the
endpoint in the $p_{2}$-constant segment, with smallest $p_{1}$-coordinate.
For that to happen, the $p_{2}$-coordinate of the red curve needs to move
different as we move along the different parts of the coamoeba, as indicated
in Figure 19. So it maintains the property that the $p_{2}$-coordinate at the
common point of the segments labeled $2$ and $3$ is smaller than the one
corresponding to the $5$ and $6$ segments. We also need to ensure the
following. Consider the point of intersection between the $(p_{1},p_{2})$
projection of the top blue curve and the projection of $S_{i}$. Look at the
$\theta_{2}$-coordinate of the circle of $S_{i}$ over this point, and consider
its intersections $q_{1}$, $q_{2}$ with the blue curve. We draw the coamoeba
profile of the red curve so that $q_{1}$, $q_{2}$ are precisely the pink
vertices, as illustrated in Figure 18 (Center). We make analogous choices for
drawing the red curves at the bottom of the coamoeba.

We are now in shape to prove the following lemma.

######  Lemma 9.

There is a symplectic disk ${\Sigma^{\prime}}_{j}^{i,i+1}$ in the cut
neighborhood $\mathcal{M}$, disjoint from the Lagrangian 2-sphere $S_{i}$,
containing the ${\sigma^{\prime}}^{i}_{j}$ and ${\sigma^{\prime}}^{i+1}_{j}$
symplectic 2-disks as a subset, with boundary in a thickened torus
$\mathbb{T}$, whose homology class is twice the collapsing class of the nodes
via the identification $H_{1}(\mathbb{T};\Z)\cong H_{1}(T^{2};\Z)$.

Denote by $\mathcal{P}^{\prime}$ the pair of pants which is the closure of
${\Sigma^{\prime}}_{j}^{i,i+1}\setminus({\sigma^{\prime}}^{i}_{j}\cup{\sigma^{\prime}}^{i+1}_{j})$.

######  Proof 4.19.

As in the previous proof, we will draw in the $(p_{1},p_{2})$-coordinates the
level sets of the $\theta_{2}$ and $\theta_{1}$ coordinates, naming the former
$\xi^{1}$ and the later $\xi^{2}$. Let us start looking at the bi-gon, and
describe the projection of the $\xi^{1}$ curve in the amoeba. Each curve
starting close to the $\theta_{2}$ minimum, up to a certain height $b$ (to be
specified), will have a horizontal projection, with the maximum of the $p_{1}$
coordinate corresponding to the half of the $\theta_{1}$ coordinate of
$\xi^{1}$. This last property remains the same, even when we start at a height
greater than $b$, but then, the image becomes a graph of a non-increasing
function $p_{2}(p_{1})$, eventually limiting to the part of the pink curve
that stops at $x$, as illustrated in the top-right picture of Figure 20. We
assume that the derivative is smaller in norm to the derivative of the graph
giving the pink curve. For the top part of the coamoeba, the $\xi^{1}$ curves
have constant $p_{2}$ coordinates, with the minimum of the $p_{1}$ coordinate
happening at the middle of the $\theta_{1}$ coordinate. The amoeba projection
of the $\xi^{2}$ curves are also indicated in top-right picture of Figure 20.
In particular, analyzing the symplectic condition for the top of the coamoeba
part, is essentially done as in the proof of Lemma 8, compare top-middle and
top-right pictures of Figure 20.

To ensure the symplectic condition at the bottom of the coamoeba, we need to
carefully choose the point $b$, recalling that we can choose the red curve so
that the $\theta_{2}$ variation is small enough compared with the $\theta_{1}$
variation. Away form the points in $\xi^{1}$ with maximum $p_{1}$ coordinate,
let us move in the $\xi^{i}$ curves with the normalised condition of
$|dp_{1}^{i}|=1$. The variation $d\theta_{2}^{2}$ then will be bounded by an
extremely small constant (w.r.t. the $\theta_{1}$ diameter the coamoeba). We
take a constant $b$, high enough to ensure that
$\omega(\partial\xi^{1},\partial\xi^{2})=d\theta_{1}^{1}-dp_{2}^{1}d\theta_{2}^{2}>0$,
recalling that we forced $dp_{2}^{1}$ to be zero for points at $\theta_{2}$
heights smaller than $b$ and $|dp_{2}^{1}|$ is bounded by the maximum slope of
the pink curve. In the points in $\xi^{1}$ with maximum $p_{1}$ coordinate, we
simply have $dp_{2}^{1}=0$ and
$\omega(\partial\xi^{1},\partial\xi^{2})=-d\theta_{1}^{1}dp_{1}^{2}>0$. We let
the reader check the positivity for the limiting points at the pink curve.

Let us now move to the tri-gon part of the coamoeba. For the top part we
choose a height $c$ in the coamoeba, playing a similar role as $b$ in the
above paragraph. If the $\theta_{2}$ coordinate of $\xi^{1}$ is bigger than
$c$, we take the amoeba part to have constant $p_{2}$ coordinate. So if the
$\theta_{2}$ coordinate is not smaller than $c$, we have $dp_{2}^{1}=0$ and
$\omega(\partial\xi^{1},\partial\xi^{2})=-d\theta_{1}^{1}dp_{1}^{2}>0$. The
analysis regarding the symplectic condition is done as in the last paragraph
for the part corresponding to the $\theta_{2}$ coordinate smaller than $c$.
For the bottom part, recall that the $p_{2}$ coordinate corresponding to the
part $2$ of the coamoeba of the red curve is smaller than the one
corresponding to the part $6$. So we can take $dp^{1}_{2}>0$. The analysis now
is similar to the analogous part in the proof of Lemma 8. We have
$d\theta^{1}_{1}\geq 0$ (being zero at the middle of the $\xi^{1}$ curves
whose $\theta_{2}$ coordinate is not bigger than the pink vertices),
$dp_{2}^{1}>0$, $d\theta^{1}_{2}=0$, $dp^{2}_{1}<0$, $d\theta_{1}^{2}=0$,
$d\theta^{2}_{2}>0$, since we move on $\xi^{2}$ from the red curve to the blue
curve.

The fact that we chose the pink curve to cross $p_{1}=0$ with negative $p_{2}$
coordinate promptly ensures that the coamoeba region of $S_{i}$ corresponding
to the segment given by the intersection of $p_{1}=0$ with amoeba of the bi-
gon, does not intersect the bi-gon itself. For the tri-gon part, looking at
the amoeba projection, we see that $S_{i}$ does not intersect the region of
the surface corresponding to the bottom of the tri-gon. The fact that we chose
the coamoeba of our red curve so that the pink vertices coincide with the
points $q_{1}$, $q_{2}$, ensures that the part of the surface whose coamoeba
correspond to top of the tri-gon does not intersect the Lagrangian $S_{i}$ as
well, as required.

There is a smooth way to isotope the boundary of $\mathcal{P}$ to the boundary
of $\mathcal{P}^{\prime}$, and following that, a smooth way to isotope the
amoebas of $\Sigma_{j}^{i,i+1}$ to the ones of $\Sigma_{j}^{i,i+1}$, see again
Figure 20, as well as the small differences on the coamoebas. Hence, we can
isotope from $\Sigma_{j}^{i,i+1}$ to $\Sigma_{j}^{i,i+1}$, with their
boundaries restricted to a thickened torus $\mathbb{T}=I\times T^{2}$. $\Box$

Now we see that the symplectic surface ${\Sigma^{\prime}}_{j}^{i,i+1}$
intersects the Lagrangian 2-sphere $S_{i+1}$ once in a point belonging to
${\sigma^{\prime}}_{j}^{i+1}$. In an analogous fashion to Lemma 9, we can chop
out of ${\Sigma^{\prime}}_{j}^{i,i+1}$ the intersection point with $S_{i+1}$
and then glue its boundary, the boundary of ${\sigma^{\prime}}_{j}^{i+2}$
(where we remove the intersection of $\sigma_{j}^{i+2}$ with $S_{i+1}$), and
the boundary of a new chosen “red curve” in a thickened torus that has three
times the homology of the collapsing cycle in this thickened torus, with a new
pair of pants. Naming the former two curves blue, the first step would be to
construct a dimer model between the blue curves and the red curve, so that the
behaviour on the region that it could intersect $S_{i+1}$ is the same as the
one analyzed in Lemma 9. Denote this surface by
${\Sigma^{\prime}}_{j}^{i,i+2}$. We can iterate this process and consider
symplectic surfaces ${\Sigma^{\prime}}_{j}^{i,i+k}$ in the cut neighborhood
$\mathcal{M}$, that do not intersect $S_{i},\dots,S_{i+k-1}$, and has boundary
on a thickened torus, whose homology class is $k+1$ times the collapsing
cycle. This process leads to the following:

######  Lemma 10.

There exists a symplectic disk ${\Sigma^{\prime}}_{j}^{1,n}$ inside
$\mathcal{M}$, not intersecting the Lagrangian set $\bigcup_{1=1}^{n-1}S_{i}$,
and whose boundary lies on a thickened torus $I\times T^{2}$, with boundary
class being $n$ times the class of the collapsing cycle.

######  Proof 4.20.

We build this surface iteratively as indicated above, starting with
${\Sigma^{\prime}}_{j}^{1,2}$. The algorithm to build the red curve and the
corresponding dimer model is illustrated for going from
${\Sigma^{\prime}}_{j}^{1,3}\cup\sigma_{j}^{4}$ to
${\Sigma^{\prime}}_{j}^{1,4}$ in the third diagram of Figure 18. We chose the
red curve to cross in consecutive chambers of the complement of the curves
given by the coamoeba projection of the blue curves, which we recall is the
boundary of the disconnected surface obtained by chopping out the
intersections of ${\Sigma^{\prime}}_{j}^{1,i}\cup\sigma_{j}^{i+1}$ with
$S_{i}$. We do it so that the top part of the dimer, corresponding to one bi-
gon and two tri-gons have the same configuration as in Lemma 9, including the
points analogous to $q_{1},q_{2}$. Recalling Remark 4.4.2, we can think that
we first glue a pair of pants $\mathcal{P}$ as before for the boundaries of
${\Sigma^{\prime}}_{j}^{1,i}\cup\sigma_{j}^{i+1}$ and then isotope to our
desired surface ${\Sigma^{\prime}}_{j}^{1,i+1}$, with the modifications
happening in the same framework as in the proof of Lemma 9.

Figure 21: Final amoeba for the symplectic disk in the case of a symplectic-
tropical curve arriving at $n=4$ nodes with multiplicity $m=5+6=11$, so $k=2$
and $d=3$. For visual purposes, we illustrate the amoeba of $k=2$ surfaces as
if one envelopes the other, when in fact both have the same amoebas.

Now we should inductively build the surfaces ${\Sigma^{\prime}}_{j}^{1,n}$,
for $j$ going from $1$ to $k$, making sure these surfaces do not intersect.
Recall that the boundaries of $\sigma_{j}$ and $\sigma_{j+1}$ are linked in
the thickened 3-dimensional neighborhood. We can achieve non-intersection by
adjusting the crossings of blue and red curves between different amoebas, and
the $p_{2}$ coordinate. Thus we can ensure the required non-intersection just
by looking at the $(\theta_{1},\theta_{2},p_{2})$ projection of the surfaces.
The case $n=2$ is illustrated in Figure 18 (d). We can then get the other $d$
disks, carrying the collapsing cycle from the respective nodes to boundary of
the same thickened torus. They project to curves inside the amoeba of the
${\Sigma^{\prime}}_{j}^{1,n}$ disks, see Figure 21. After doing that, still
within the cut neighborhood $\mathcal{M}$, we can connect the boundaries of
the ${\Sigma^{\prime}}_{j}^{1,n}$ surfaces, $j=1,\dots,k$ and of the $d$ disks
into $m=kn+d$ straight cycles and redistribute these cycles over curves
connecting to our symplectic-tropical curve $\mathscr{C}(\Gamma)$, as we did
in the Section 4.3.3. Figure 21 illustrates the amoeba image of this local
model of a deformed symplectic-tropical curve, when $n=4$, $m=11$, so $k=2$
and $d=3$. This finishes the proof of Proposition 4.17. ∎

We can glue all these local models now, as we did in Section 4.3.3, to get
deformed symplectic-tropical curves, intersecting Lagrangian spheres only at
prescribed nodes, with the number determined by the total multiplicity and the
number of nodes at a given cut.

###  4.5 Required Symplectic-Tropical Curves in the $\mathfrak{N}$ for Theorem
1

Consider the triangular-shaped ATFs of $\mathbb{CP}^{2}$,
$\mathbb{CP}^{1}\times\mathbb{CP}^{1}$, $Bl_{3}(\mathbb{CP}^{2})$ and
$Bl_{4}(\mathbb{CP}^{2})$, [55] with one smooth corner, i.e., satisfying a
Markov type equation of the form

(4.8) $1+n_{2}q^{2}+n_{3}r^{2}=Gqr.$

Let $\mathfrak{N}$ be a neighborhood of the edge opposite the frozen smooth
vertex, introduced in Section 4.4.1. The results of the previous subsections
yield the following

######  Theorem 11.

Each homology class of the symplectic divisors highlighted in Figure 1 can be
realized as a symplectic-tropical curve in $\mathfrak{N}$.

######  Proof 4.21.

First note that for the equations associated with $\mathbb{CP}^{2}$,
$\mathbb{CP}^{1}\times\mathbb{CP}^{1}$, $Bl_{3}(\mathbb{CP}^{2})$ and
$Bl_{4}(\mathbb{CP}^{2})$, the quantity $G/n_{2}n_{3}$ is respectively, $3$,
$2$, $1$, $1$, which is the intersection number of the symplectic divisors in
the corresponding spaces with the anti-canonical divisor. Thus, Theorem 7 and
Section 4.4.1 suffice for the case of $\mathbb{CP}^{2}$. For the remaining
cases, one needs to ensure the correct intersection with the Lagrangian
spheres. This can be achieved case by case using that

$1+n_{2}q^{2}\equiv 0\mod n_{3},\qquad 1+n_{3}r^{2}\equiv 0\mod n_{2}.$

For instance, in the case of $Bl_{4}(\mathbb{CP}^{2})$, we get
$q^{2}\equiv-1\mod 5$, so $q\equiv 2$, or $3\mod 5$. Take the divisor of
Figure 1 that intersects two Lagrangian spheres. Following the mutations in
[55, Figure 17], at the triangular-shape ATBD [55, Figure 17 ($A_{4}$)], these
spheres become the top and bottom spheres, of the 4-chain of Lagrangian
spheres. If $q\equiv 2\mod 5$, apply Proposition 4.17 taking the 2
intersections to the top and bottom node. If $q\equiv 3\mod 5$, we arrive at
the three interior nodes instead, having the same intersection number with the
Lagrangian spheres. One can check that the sign is correct by looking at the
first instance when $q=2$ and observe that the mutations $q\to 5r-q$ switch
between $q\equiv 2\mod 5$ and $q\equiv 3\mod 5$ (the first case $q=2$ is
depicted as $A_{1}$ in Figure 23).

###  4.6 Further Deformations of Symplectic Tropical Curves

In this subection we introduce a series of additional techniques regarding
symplectic-tropical curves, that will allow us to visualize chains of them
inside an ATF. When we say a chain of symplectic curves, we imply that the
total intersection between them is equal to the geometric intersection. Thus,
it is not enough to simply construct, for each curve in the chain a STC, as we
did in the previous sections, as we want to geometrically realize the
homological intersection. We start with a simple observation:

First, for $i=1,2$, let $\mathscr{C}_{i}$ denote two STCs as in Definition
4.6. Let $C_{i}$ denote a STC in $X$ represented by $\mathscr{C}_{i}$ as in
Theorem 7, and let $\gamma_{i}$ be an edge of $\mathscr{C}_{i}$. If
$\gamma_{1}$ and $\gamma_{2}$ homology class of the torus, any intersection
between $\mathscr{C}_{1}(\gamma_{1})$ and $\mathscr{C}_{2}(\gamma_{2})$, can
be taken to be empty as an intersection of the symplectic surface $C_{1}$ and
$C_{2}$, since we can just assume we carry disjoint cycles in the same
homology class. $\Box$

Figure 22: (a) The amoeba of a STC given by a $\sigma_{j}$ disk (from
Proposition 4.9), glued along the boundary with two cycles intersecting once
$\partial\sigma_{j}$; (b) The coamoeba of the gluing surface; (c) A depiction
of how the surface looks like inside a thickened torus.

We will also need the following:

######  Proposition 4.22.

Consider a thickened torus $\mathbb{T}=[-\epsilon,\epsilon]\times T^{2}$, as
the pre-image of a segment in the regular part of a base of an ATF. Let
$\alpha$ be a straight cycle in $\\{0\\}\times T^{2}$ represented by $v\in
H_{1}(\mathbb{T};\Z)\cong H_{1}(T^{2};\Z)\cong\Z^{2}$, $\beta$ be a cycle in
the class $v$ that wraps around once $\alpha$, and $\gamma_{\pm}$ be a
straight cycle in $\\{\pm\epsilon\\}\times T^{2}$, represented by $u_{\pm}$,
with $\det u\wedge v=\pm 1$, and $u_{-}=u_{+}-v$.

Then there exists a symplectic pair of pants in the complement in
$\mathbb{T}\setminus\alpha$, with boundary
$\gamma_{-}\cup\beta\cup\gamma_{+}$.

######  Proof 4.23.

Use the dimer model represented in Figure 22 (b) to build a symplectic surface
as in Section 4.3.2, making sure that the $0$ level set of the height function
($\rho_{2}$ in Proposition 4.14) is disjoint from the straight cycle $\alpha$.
The end result is depicted in Figure 22 (c).

Applying this result for $\partial\sigma_{j}$, the boundary of a disk
$\sigma_{j}$ as in Proposition 4.9, we see that we can pass with all
$\sigma_{l}$, with $l<j$ (the ones with boundary closer to
$\alpha=\partial\sigma_{1}$) through the middle of the surface constructed in
Proposition 4.22. The amoeba of this process is depicted in Figure 22 (a).

Figure 23: STCs in classes $H_{1}$, $H_{2}$ in a specific ATBD of
$\mathbb{CP}^{1}\times\mathbb{CP}^{1}$, in the exceptional classes $E_{1}$,
$E_{2}$, $B_{2}=H-E_{1}-E_{3}$ and $B_{3}=H-E_{1}-E_{2}$ in an ATBD of
$Bl_{3}(\mathbb{CP}^{2})$, and in the exceptional classes
$A_{1}=H-E_{1}-E_{4}$, $A_{2}=E_{4}$, and $A_{5}=E_{1}$ in an ATBD of
$Bl_{4}(\mathbb{CP}^{2})$.

In Figure 23, we apply the above Remark 4.6 to visualise STCs in the
neighborhood $\mathfrak{N}$ of highlighted edge of the ATF of
$\mathbb{CP}^{1}\times\mathbb{CP}^{1}$ depicted in the top corner of Figure 1.
They are representative of the classes
$H_{1}=[\C\mathbb{P}^{1}\times\\{\text{point}\\}]$ and
$H_{2}=[\\{\text{point}\\}\times\C\mathbb{P}^{1}]$. Remarks 4.6, 4.6, are used
to visualize a 4-chain of STCs in the neighborhood $\mathfrak{N}$ of
highlighted edge of the ATF of $Bl_{3}(\mathbb{CP}^{2})$ of triangular shape
depicted in Figure 1. The homology classes for the spheres in this 4-chain are
the ones corresponding to the highlighted edges of the toric diagram for
$Bl_{3}(\mathbb{CP}^{2})$ in Figure 1. Figure 23 also shows a 3-chain of
symplectic spheres in the ATBD of $Bl_{4}(\mathbb{CP}^{2})$ of [55, Diagram
$(A_{4})$]. Their classes corresponds to the highlighted 3-chain in the first
diagram of $Bl_{4}(\mathbb{CP}^{2})$ in Figure 1.

Now, we can iteratively apply Remark 4.6 in the neighborhood of one or more
nodes. We are going to use simplified pictures, for visual purposes. For
instance, Figure 24 (a) shows a simplified depiction of two nonintersecting
STCs near two nodes of a cut in an ATF, with associated vector $v$.
Surrounding each node, we see $a$ $\sigma_{j}$ type curve, where we applied
Remark 4.6 $a$ times around each node and then unite their amoebas using
Remark 4.6. Figure 24 (b) shows an alternative version, where we took $2b$
curves around a unique node in associated with the vector $w$, and applied
Remarks 4.6, 4.6 to get two disjoint symplectic curves.

Figure 24: Simplified pictures for the STC discussed above.

Let us now shift our focus to what happens as we approach the local model
nearby a trivalent vertice of an STC, with two (or more) sets of curves
satisfying the balancing condition (4.1). The first observation is that if you
arrive with two sets of cycles with total homology represented by
$\emph{{v}}\in\R^{2}$ and want to glue them to sets of cycles with homoloy w
and $-\emph{{w}}-\emph{{v}}$, using two non-intersecting surfaces, the price
you pay is that the cycles corresponding to w and $-\emph{{w}}-\emph{{v}}$ in
the boundary of the second surface, must link the corresponding boundaries on
the first surface a total amount444For us $\emph{{w}}\wedge\emph{{v}}$ denotes
the determinant of the two vectors in $\R^{2}$. of
$\emph{{w}}\wedge\emph{{v}}$. We define precisely what we mean by cycles
linking within the regular part of a Lagrangian torus fibration:

######  Definition 4.24.

Let $\mathbb{D}$ be a 2-disk, fix a point $o\in\mathbb{D}$ and take two
disjoint 1-cycles $\alpha$, $\sigma$ in $\mathbb{D}\times T^{2}$ away from
$\\{o\\}\times T^{2}$. We view $\sigma$ as a cycle in $H_{1}(\mathbb{D}\times
T^{2}\setminus\alpha;\Z)\cong H_{1}(T^{2};\Z)\oplus\Z$, where the first
summand corresponds to $\\{o\\}\times T^{2}\hookrightarrow\mathbb{D}\times
T^{2}\setminus\alpha$.

By definition, the _linking_ between $\sigma$ and $\alpha$ is the projection
of the class $[\sigma]\in H_{1}(\mathbb{D}\times T^{2}\setminus\alpha;\Z)$
onto the (rightmost) $\Z$-factor. The sign involves a choice of generator for
the $\Z$ cycle, that we assume the same, when dealing with more than one
1-cycle relative to $\alpha$. $\Box$

Let us summarize the above discussion into the following statement:

######  Proposition 4.25.

Consider the local model for a symplectic surface near the interior vertex
constructed in Proposition 4.14, associated to the balancing condition
$m_{1}\emph{{w}}_{1}+m_{2}\emph{{w}}_{2}+m_{3}\emph{{w}}_{3}=0$. Consider the
number $d=m_{1}m_{2}|\emph{{w}}_{1}\wedge\emph{{w}}_{2}|$ and
$d=\delta_{1}+\delta_{2}$, a two partition $\delta_{1},\delta_{2}\in\Z_{\geq
0}$.

Then there exists another disjoint symplectic surface in the same local
neighborhood such that the boundaries satisfies the following two conditions:

  * -

The $m_{3}$ boundaries associated to $\emph{{w}}_{3}$ are parallel copies of
the corresponding boundaries of the original curve inside the same torus
fibre;

  * -

The boundaries associated to $\emph{{w}}_{1}$ and $\emph{{w}}_{2}$ link the
corresponding boundaries of the original curve, in the sense of Definition
4.24, $\delta_{1}$ and $\delta_{2}$ times, respectively.

######  Proof 4.26.

We start with the dimer model for the original surface constructed in
Proposition 4.14. Recall that we used $(\rho_{1},\rho_{2})$ coordinates for
the amoeba description, with $\rho_{2}\in[-\epsilon,\epsilon]$. We will
construct another dimer model for our second surface, and build the amoeba as
in Sections 4.3.2, 4.3.3, 4.4.2.

Figure 25: Near an interior vertice of determinant $d$, there are disjoint
symplectic curves with the total linking of their boundaries equal to $d$.
This series of diagrams illustrate the algorithm to get the disjoint surfaces.

In the intersection of these two dimer models we will record the
$\rho_{2}$-coordinate of the new curve in the dimer model, to be less than
$-\epsilon$ or greater than $\epsilon$. This will be indicated in the same
diagram, as follows. If coamoeba regions of the two surfaces intersect, the
boundary of the lower region will be denoted by a dotted segment in the
intersection. Figure 25 (d) illustrates two non-intersecting surfaces via
their coamoeba projection. By the work developed in Sections 4.3.2, 4.3.3,
4.4.2, the construction of such dimer model, with the additional $\rho_{2}$
information, will be enough to ensure that we obtain two disjoint symplectic
surfaces.

As before we take $\emph{{w}}_{1}=(1,0)$, and we use the following algorithm
to construct the dimer model, which concludes Proposition 4.25.

Algorithm for the Dimer Model:

  1. Step 1

Color the $\emph{{w}}_{1}$ cycles red, $\emph{{w}}_{2}$ cycles blue,
$\emph{{w}}_{3}$ cycles green.

  2. Step 2

For each of the $|d|=m_{1}m_{2}|\emph{{w}}_{1}\wedge\emph{{w}}_{2}|$
intersections between the red and blue cycles of the original curve, draw
cycles linking both blue and red cycles, as illustrated in Figure 25 (a), in
the same pattern. Color $\delta_{1}$ of them red and the other $\delta_{2}$ of
them blue.

  3. Step 3

Consider a new green cycle, parallel to the original green, constructed as a
positive shift in, say, the $\phi_{2}$ coordinate of the amoeba (the
$\phi_{i}$-coordinates being the coamoeba coordinates as in the notation of
Proposition 4.14).

  4. Step 4

Replace the red/blue links of Step 2, by chains of the same color, “linking”
the corresponding original curve, with one end on the new green cycle and
crossing the original blue and red cycle twice, to the left of the new green
cycle. The first crossing is above and the second below the original dimer,
with respect to the $\rho_{2}$ coordinate, as illustrated in Figure 25 (b).

  5. Step 5

For each red/blue linking chain, we connect its “tail” with the “head” of the
adjacent chain of the same color, using a red/blue 1-chain parallel to the
original chain of the same color, forming the new red/blue cycles as
illustrated in Figure 25 (c).

  6. Step 6

Paint the regions that were created by the new green, red and blue cycles,
passing below or above the original dimer accordingly, as illustrated in
Figure 25 (d).

######  Remark 4.27.

We could allow linkings for the $\emph{{w}}_{3}$ cycles, provided the total
linking is still $d$. This is not needed for our purposes, and it would make
the construction algorithm more intricate. $\Box$

######  Remark 4.28.

After getting the two surfaces of Proposition 4.25, one can actually run an
analogous algorithm to get yet a third, fourth and $n$-th surfaces, disjoint
from the previous ones, with boundary so that the green cycle is parallel to
the previous green cycles, and the red/blue cycles links each red/blue cycles
of the previous surfaces $\delta_{1}$/$\delta_{2}$ times. For that, one just
needs to replicate the intersection pattern of the $n$-th red/blue curve, with
the previous $(n-1)$ ones. $\Box$

Now assume that we arrive at the surface near the interior node, constructed
in Proposition 4.14, with $m_{3}$ straight (green) $\emph{{w}}_{3}$ cycles,
parallel to the original one, and $m_{1}$ (red) $\emph{{w}}_{1}$ cycle,
linking the original curve $c$ times. We can adjust the new red curves, so
that the curves arrive linking the dimer model over one red/blue vertex as in
Figure 26 (a).

######  Proposition 4.29.

Considering the above setup and $d\in\N$ as in Proposition 4.25. Then there
exists a symplectic surface connecting the $m_{3}$ $\emph{{w}}_{3}$-cycles
(green) and $m_{1}$ $\emph{{w}}_{1}$-cycles (red), with $m_{2}$
$\emph{{w}}_{2}$-cycles (blue), linking the original $\emph{{w}}_{2}$-cycles
$(c+d)$ times.

Figure 26: How linking can change for non-intersecting symplectic surfaces
near the interior vertex of a STC. Picture (b) is a zoomed version around the
node with red links. The blue curve is a bit rotated for visual purposes.

######  Proof 4.30.

At each vertex having the linking of the red curves, we construct a local
model as illustrated by Figure 26 (a) and (b). We note that the new blue chain
links the original blue at the red linking number plus 1. In Figure 26, the
red cycles are linking 4 times and the new blue cycle links 5 times the
original blue cycle. (A generalization of the picture is clear.) Now we add
blue links to the remaining intersections of the originals blue and red
curves, as in Step 2 of the algorithm of Proposition 4.25. Then we can run
Step 5, Step 6 of the above algorithm in an analogous fashion, noting that the
local model of Figure 26 (b) is well adapted for that.

As in Remark 4.28, if we assume that we arrive with another set of $m_{1}$ red
cycles linking the previous ones $c$ times, and a set of $m_{3}$ parallel
green cycles, we can – after an initial adjustment, concentrating the several
red links in one node as before – locally construct the new blue link, linking
both previous blue link equal to the local red linking number plus one. Close
to the boundary of the local region, the third local surface will lay above
the previous two. This can also be iterated $n$ times, so that the local
behaviour glues well with the $n$-th iteration of the algorithm, described in
Remark 4.28. $\Box$

###  4.7 Getting chains of symplectic-tropical curves

Let us apply the results of Section 4.6 to construct the required chains of
symplectic tropical curves used in Theorem 3. There are three cases,
corresponding to $\mathbb{CP}^{1}\times\mathbb{CP}^{1}$,
$Bl_{3}(\mathbb{CP}^{2})$ and $Bl_{4}(\mathbb{CP}^{2})$, which we now analyze.

####  4.7.1 The case of $\mathbb{CP}^{1}\times\mathbb{CP}^{1}$.

Consider the triangular-shaped ATF of the symplectic monotone
$\mathbb{CP}^{1}\times\mathbb{CP}^{1}$, with a smooth corner, associated to a
solution of the Diophantine equation

$1+q^{2}+2r^{2}=4qr.$

Let $\mathfrak{N}$ be a neighborhood of the edge opposite the smooth corner –
where the frozen vertex is located – and consider its associated cuts, as in
Section 4.4.1.

######  Proposition 4.31.

There is a 2-chain of symplectic-tropical curves inside the edge neighborhood
$\mathfrak{N}$, such that the associated symplectic curves belong to the
classes $H_{1}$ and $H_{2}$, and have total intersection number one, i.e.,
equal to their topological intersection.

######  Proof 4.32.

For that, we need to revisit the specific combinatorics in this situation. In
this case, the associated Markov type equation of interest is
$1+q^{2}+2r^{2}=4qr.$ From Equation (4.5), we see that the determinant between
the associated vectors $v=(-r,m)$ and $w=(q,-l)$ is $v\wedge w=2$. It follows
from the corresponding Vieta jumping (Proposition 3.8) that $q=2a+1$ and
$r=2b+1$ are odd. Hence, we can rewrite the balancing condition as:

(4.9) $qv+rw+2(0,1)=v+w+2u=0$

where $u=av+bw+(0,1)$. We readily see that $w\wedge u=u\wedge v=1$. In
consequence, we are allowed to use the vectors $\pm u$, with each of the
vectors $v$ and $w$ as in Proposition 4.22. Finally, we look at

$u\wedge(0,1)=-\frac{q-1}{2}r+\frac{r-1}{2}q=\frac{r-q}{2}.$

Assume that $q>r$, so that $u$ points to the left. Then we deduce that

$2v-u\wedge(0,1)=\frac{q-5r}{2}<0,$

because $1+2r^{2}=q(4r-q)$, so $q<4r$.

Figure 27: STCs in classes $H_{1}$, $H_{2}$ in a neighborhood $\mathfrak{N}$
of the edge opposite to the smooth corner in a arbitrary ATF of
$\mathbb{CP}^{1}\times\mathbb{CP}^{1}$ associated with a solution of
$1+q^{2}+2r^{2}=4qr$.

Now, Figure 27 is a depiction of the chain of symplectic spheres in
$\mathfrak{N}$ in classes $H_{1}$ and $H_{2}$. The intersection is depicted by
a star. The top picture records the homology classes of the cycles
corresponding to each edge. The bottom picture records the linking between the
two cycles obtained by applying Propositions 4.25 and 4.29. We note that we
get the Markov equation as a compatibility equation for the interior vertice
associated with the $2(0,1)$ cycles. Indeed, since $qv\wedge rw=2qr$, we must
have

$2qr=r^{2}+a+1+aq=\frac{2r^{2}+q-1+2+(q-1)q}{2}=\frac{2r^{2}+q^{2}+1}{2}.$

In the case $r>q$, we replace the vector $u$ by $-u$ in Figure 27, and note
that $u\wedge(0,1)>0$ and

$2v+u\wedge(0,1)=\frac{-q-3r}{2}<0.$

The case $r=q$ is depicted in Figure 23.

######  Remark 4.33.

The construction obtained by the above picture in Figure 27 is equivalent to
the construction of two (geometrically) disjoint copies of the $H_{1}$ class,
to which we can apply a Dehn twist with respect to the visible Lagrangian
2-sphere. $\Box$

####  4.7.2 The case of $Bl_{3}(\mathbb{CP}^{2})$.

Let us now consider a triangular-shaped ATF for the symplectic 4-manifold
$Bl_{3}(\mathbb{CP}^{2})$, with a smooth corner, associated to a solution of
the Diophantine equation

$1+2q^{2}+3r^{2}=6qr,$

$\mathfrak{N}$ a neighborhood of the edge opposite the smooth corner, and its
associated cuts, as in Section 4.4.1. In the case, the required chain of
symplectic surfaces reads:

######  Proposition 4.34.

There exists a 4-chain of symplectic-tropical curves inside the edge
neighborhood $\mathfrak{N}$ whose associated symplectic curves lie in the
exceptional classes $E_{1}$, $E_{2}$, $B_{2}=H-E_{1}-E_{3}$ and
$B_{3}=H-E_{1}-E_{2}$, and intersection between two of them equals their
geometric intersection.

######  Proof 4.35.

From Equation (4.5), we deduce that the determinant between associated vectors
$v=(-r,m)$ and $w=(q,-l)$ is $v\wedge w=1$. From the corresponding Vieta
jumping (Proposition 3.8), we obtain that $q^{2}\equiv-1\mod 3$, and thus
$q\equiv 1$ or $2\mod 3$, and $r\equiv 1\mod 2$.

Figure 28: STCs in classes $E_{1}$, $E_{2}$, $B_{2}$ and $B_{3}$ in a
neighborhood $\mathfrak{N}$ of the edge opposite to the smooth corner in a
arbitrary ATF of $Bl_{3}(\mathbb{CP}^{2})$ associated with a solution of
$1+2q^{2}+3r^{2}=6qr$.

Figure 28 illustrates the case $q=3a+1$, $r=2b+1$. As before, the intersection
is depicted by a star in the bottom picture, and occurs exactly as we change
the linking number between cycles. The top picture records the homology
classes of the cycles corresponding to each edge. The bottom picture records
the linking between the two cycles by applying Propositions 4.25 and 4.29. In
this case, the linking numbers $\delta$ and $\varepsilon$, will depend on the
curves we are taking into account. We have that
$\varepsilon,\delta\in\\{0,1\\}$ and $\delta+\varepsilon=1$. The compatibility
condition becomes:

$qr=b+\varepsilon+br+a+\delta+aq=\frac{3(r-1)(r+1)+6+2(q-1)(q+1)}{6}=\frac{3r^{2}+2q^{2}+1}{6}$

Figure 29 illustrates the second case $q=3a+2$, $r=2b+1$. In this case, we
have $\varepsilon\in\\{0,1\\}$, $\delta\in\\{1,2\\}$ and
$\delta+\varepsilon=2$, where the compatibility becomes:

$qr=b+\varepsilon+br+2a+\delta+aq=\frac{3(r-1)(r+1)+12+2(q-2)(q+2)}{6}=\frac{3r^{2}+2q^{2}+1}{6},$

as required. This concludes the construction for the case of
$Bl_{3}(\mathbb{CP}^{2})$.

Figure 29: STCs in classes $E_{1}$, $E_{2}$, $B_{2}$ and $B_{3}$ in a
neighborhood $\mathfrak{N}$ of the edge opposite to the smooth corner in a
arbitrary ATF of $Bl_{3}(\mathbb{CP}^{2})$ associated with a solution of
$1+2q^{2}+3r^{2}=6qr$.

####  4.7.3 The case of $Bl_{4}(\mathbb{CP}^{2})$.

Finally, we consider a triangular-shaped ATF for the symplectic surface
$Bl_{4}(\mathbb{CP}^{2})$, with a smooth corner, associated to a solution of
the Diophantine equation

$1+q^{2}+5r^{2}=5qr,$

$\mathfrak{N}$ a neighborhood of the edge opposite the smooth corner, together
with the associated cuts, as in Section 4.4.1. The required chain of
symplectic curves is obtained in the following

Figure 30: STCs in classes $A_{1}$, $A_{2}$ and $A_{5}$ in a neighborhood
$\mathfrak{N}$ of the edge opposite to the smooth corner in a arbitrary ATF of
$Bl_{4}(\mathbb{CP}^{2})$ associated with a solution of $1+q^{2}+5r^{2}=5qr$.

######  Proposition 4.36.

There exists a 3-chain of symplectic-tropical curves inside the edge
neighborhood $\mathfrak{N}$ whose associated symplectic curves belong to the
exceptional classes $A_{1}=H-E_{1}-E_{4}$, $A_{2}=E_{4}$, and $A_{5}=E_{1}$,
and their pairwise intersections equal their geometric intersections.

######  Proof 4.37.

Let us revisit the specific combinatorics of the situation: the associated
Markov type equation of interest is $1+q^{2}+5r^{2}=5qr$, and from Equation
(4.5), the determinant between the associated vectors $v=(-r,m)$ and
$w=(q,-l)$ is $v\wedge w=1$. As above, Proposition 3.8 shows that
$q^{2}\equiv-1\mod 5$, and thus $q\equiv 2$ or $3\mod 5$. Figure 30 shows the
case $q=2+5a$.

Figure 31: STCs in classes $A_{1}$, $A_{2}$ and $A_{5}$ in a neighborhood
$\mathfrak{N}$ of the edge opposite to the smooth corner in a arbitrary ATF of
$Bl_{4}(\mathbb{CP}^{2})$ associated with a solution of $1+q^{2}+5r^{2}=5qr$.

As before, we have the compatibility associated with the $(0,1)$ cycles, using
$qv\wedge rw=qr$, and giving the associated Diophantine equation:

$qr=r^{2}+2a+1+aq=\frac{5r^{2}+(q-2)(q+2)+5}{5}=\frac{5r^{2}+q^{2}+1}{5}.$

For the second case, Figure 31 shows the case $q=3+5a$, and the compatibility
becomes:

$qr=r^{2}+3a+2+aq=\frac{5r^{2}+(q-3)(q+3)+10}{5}=\frac{5r^{2}+q^{2}+1}{5}.$

This concludes the verification for the case of $Bl_{4}(\mathbb{CP}^{2})$.

##  5 Quiver Combinatorics and Ellipsoid Embeddings

The aim of this section is to discuss the connection between the numerics of
sharp ellipsoid embeddings and the arithmetic of cluster algebras. Let
$M\cong\Z^{2}$ be the character lattice of the algebraic torus $(\C^{*})^{2}$,
and $N=\Hom(M,\Z)$ its dual lattice. Let us fix a 4-dimensional symplectic
toric variety $X_{\Delta}=(X,\omega)$ with fan $\Delta\subseteq N$, up to
$\operatorname{GL}_{2}(\Z)$-equivalence. Following the strategy presented in
this article, the first question we are addressing is the existence of a sharp
symplectic embedding

$i:E(1,a)\longrightarrow(X,\omega),$

for values of $a\in\R^{+}$. For the cases in which the symplectic embedding
$i$ exists, the second question is addressing the existence of a symplectic
embedding

$\iota:E(1,a)\longrightarrow(X\setminus D,\omega),\quad a\in\R^{+},$

where $D\subseteq(X,\omega)$ is a symplectic divisor. This second part can be
solved as in Subsection 3.3, and the arithmetic coincides for the sequence of
absolute sharp embeddings $\\{i_{n}\\}$ and its relative $\\{\iota_{n}\\}$
counterpart. Thus onwards, only the problem of ellipsoid embeddings into
closed toric varieties is addressed. Let us now provide an algebraic recipe
for potentially finding convergent sequences $\\{\alpha_{n}\\}_{n}\in\N$ of
positive real numbers such that $E(1,\alpha_{n})$ sharply embeds into
$(X,\omega)$, constructing sharp points for an infinite staircase.

For that, let us introduce a cluster algebra $C_{\Delta}$ associated to the
fan $\Delta\subseteq N$ [1, 29]. Let $E=|E(\Delta)|$ be the number of edges of
the fan $\Delta$. The initial seed for $C_{\Delta}$ is given by the following
quiver $Q_{\Delta}$:

  * 1.

The vertices of $Q_{\Delta}$ are in bijection with the edges
$\\{e_{1},\ldots,e_{E}\\}$ of the fan $\Delta$,

  * 2.

The number of arrows between to vertices $v_{i},v_{j}$, associated to edges
$e_{i},e_{j}$ is given by

$\max\\{\rho_{i}\wedge\rho_{j},0\\}$

where $\rho_{i}$ is the primitive normal vector of the edge $e_{i}$, $1\leq
i,j\leq E$.

Thus, the cluster variables for this seed are indexed by the edges
$\\{e_{1},\ldots,e_{E}\\}$ of the fan, and the skew-symmetric exchange matrix
$B$ is given by the exterior products of the two primitive normal vectors
associated to each pair of edges. Cluster mutation consists of quiver mutation
in pair with a mutation of the cluster variables. By definition, the cluster
algebra $C_{\Delta}$ associated to $\Delta$ is generated by all the cluster
variables of all the seeds obtained by mutating our initial seed.

###  5.1 The algebraic recipe

The fan associated to an ellipsoid, or rather its compactification to a
weighted projective space, is triangular. Given that the problem we address is
that of ellipsoid embeddings, the first ingredient in our recipe is a sequence
$\mu_{1},\ldots,\mu_{s}$ of mutations which brings $\Delta$ to a triangular
fan $\Delta_{0}$. These mutations are specified by an ordered choice of
possibly repeated vertices $v_{1},\ldots,v_{s}$, $v_{i}\in V(Q)$, $1\leq i\leq
s$, in the quiver. Given that the fan $\mu_{s}(\ldots(\mu_{1}(\Delta)))$ is
triangular, the sequence of mutations $\mu_{1},\ldots,\mu_{s}$ specifies a
partition of $V(Q)$ into three sets $V(Q)=T_{1}\cup T_{2}\cup F$, where
$T_{1},T_{2}$ are determined by which edge of
$\mu_{s}(\ldots(\mu_{1}(\Delta)))$ they are sent to and the set $F$ consists
of the set of vertices which are not mutated at in the list
$\mu_{1},\ldots,\mu_{s}$.

Given the quiver $Q$, declared the subset $F\subseteq V(Q_{\Delta})$ to be
frozen vertices, i.e. coefficient variables for the associated cluster
algebra. Evaluate the cluster variables in $T_{1}\cup T_{2}$ to their initial
value for a minimal Markov triple $(1,q,r)$, $q,r\in\N$, where the cluster
variables for $T_{1}$ are set to $q$ and those of $T_{2}$ to $r$. The recipe
now starts from $\mu_{s}(\ldots(\mu_{1}(Q)))$ and reads as follows:

  * (i)

Mutate at all the vertices $v\in T_{1}$,

  * (ii)

Mutate at all the vertices $v\in T_{2}$,

  * (iii)

Repeat Steps ($i$) and ($ii$).

First, it is relevant to observe that the order in which you mutate the
vertices in a given set $T_{1}$, or $T_{2}$, does not matter. This is
algebraically interesting, given that these vertices are not necessarily
disjoint, but geometrically clear.

Second, at the $k$th iteration of the recipe, $k\in\N$, the cluster variables
associated to the vertices $v\in T_{1}$ have all the same value $T_{1}(k)$,
and the cluster variables associated to the vertices $v\in T_{2}$ have all the
same value $T_{2}(k)$. The required sequence is given by

$a_{n}=T_{1}\left(\frac{n+1}{2}\right),\quad\mbox{if $n$ odd},\qquad
a_{n}=T_{2}\left(\frac{n}{2}\right),\quad\mbox{if $n$ even},$

and the infinite stair case has sharp ellipsoid embeddings points at
$\alpha_{n}$ being the appropriate Maslov successive quotients of
$\\{a_{n}\\}$. This yields a cluster algebra perspective to the numerics in
Subsection 3.2. Let us give the necessary detail for the almost-toric basis in
the second row of Figure 1.

######  Example 5.1.

Let $(X,\omega)=X(\mathscr{P})=Bl_{3}(\C\mathbb{P}^{2})$ be the monotone
delPezzo surface $dP_{3}$, corresponding to the symplectic (almost)-toric
4-manifold represented by the leftmost almost-toric base in the second row of
Figure 1. The toric fan $\Delta_{dP_{3}}$ associated to this moment polytope
is generated by the vertices $\\{(1,0),(0,1),(-1,0),(0,-1),(-1,1),(1,-1)\\}$.
The quiver $Q_{\Delta}$ is depicted in Figure 32.

Figure 32: Quiver associated to the fan of $Bl_{3}(\C\mathbb{P}^{2})$.

Let $\mu_{1},\mu_{2},\mu_{3}$ be the ordered mutations at the vertices
$v_{1},v_{5}$ and $v_{2}$, so $s=3$ in this example. The toric fan
$\Delta_{dP_{3}}$ becomes a triangular fan after these three mutations.
Indeed, the fan $\mu_{3}(\mu_{2}(\mu_{1}(\Delta_{dP_{3}})))$ is that of
$\mathbb{P}(1,2,3)$. The partition of the vertex set $Q_{\Delta_{dP_{3}}}$ is:

$T_{1}=\\{3,5\\},\quad T_{2}=\\{1,2,4\\},\quad F=\\{6\\}.$

Let us now follow the algebraic recipe above. First, initialize the six
cluster variables

$x_{1}=x_{2}=x_{3}=x_{4}=x_{5}=x_{6}=1.$

In what follows, we write the mutations for the cluster variables without
specifying the mutation in the quiver, which is assumed to occur as well in
each mutation. For instance, Figure 33 draws the quiver after mutating at the
variables $x_{1},x_{5}$ and $x_{2}$, in this order.

Figure 33: Quiver associated the fan of the symplectic toric 4-manifold
$Bl_{3}(\C\mathbb{P}^{2})$ after applying the three mutations
$\mu_{1},\mu_{2},\mu_{3}$ at vertices $v_{1}$, $v_{5}$ and $v_{2}$
respectively.

The first three mutations are at $x_{1},x_{5}$ and $x_{2}$, with resulting
cluster variables (still) being:

$\left\\{\begin{array}[]{lr}x_{1}=1,&x_{3}=1\\\ x_{2}=1,&x_{5}=1\\\
x_{4}=1,&x_{6}=1\end{array}\right\\},$

but the current seed is that associated to the quiver in Figure 33. In this
precise seed, we start mutating according to the steps. First, we mutate at
$x_{1},x_{2},x_{4}$, and we obtain:

$\left\\{\begin{array}[]{lr}x_{1}=1,&x_{3}=1\\\ x_{2}=1,&x_{5}=1\\\
x_{4}=1,&x_{6}=1\end{array}\right\\},$

Then we mutate at the variables $x_{3},x_{5}$, which leads to

$\left\\{\begin{array}[]{lr}x_{1}=1,&x_{3}=2\\\ x_{2}=1,&x_{5}=2\\\
x_{4}=1,&x_{6}=1\end{array}\right\\},$

Mutating at $x_{1},x_{2},x_{4}$ sets the cluster variables to

$\left\\{\begin{array}[]{lr}x_{1}=1,&x_{3}=2\\\ x_{2}=3,&x_{5}=2\\\
x_{4}=3,&x_{6}=1\end{array}\right\\}.$

The next few numerical seeds of these iterative process are:

$\left\\{\begin{array}[]{lr}x_{1}=3,&x_{3}=7\\\ x_{2}=3,&x_{5}=7\\\
x_{4}=3,&x_{6}=1\end{array}\right\\}\longmapsto\left\\{\begin{array}[]{lr}x_{1}=11,&x_{3}=7\\\
x_{2}=11,&x_{5}=7\\\
x_{4}=11,&x_{6}=1\end{array}\right\\}\longmapsto\left\\{\begin{array}[]{lr}x_{1}=11,&x_{3}=26\\\
x_{2}=11,&x_{5}=26\\\
x_{4}=11,&x_{6}=1\end{array}\right\\}\longmapsto\left\\{\begin{array}[]{lr}x_{1}=41,&x_{3}=26\\\
x_{2}=41,&x_{5}=26\\\ x_{4}=41,&x_{6}=1\end{array}\right\\},$

and the first few values of the sequence $a_{n}$ are:

$\\{a_{n}\\}=\\{1,1,1,2,3,7,11,26,41,97,153,362,571,1351,2131,5042,7953,18817,29681,\ldots\\}.$

The cluster variable mutation is dictated by the transformation:

$x_{n+3}=\frac{x_{n+1}x_{n+2+1}}{x_{n}},$

and satisfies the linear recurrence $x_{n}=4x_{n-2}-x_{n-4}$. This is Sequence
A005246 in the Sloane notation. Consider the Maslov weighted sequence

$A_{n}=\left\\{\begin{array}[]{lr}A_{n}=2a_{n}^{2},&n\mbox{ odd},\\\
A_{n}=3a_{n}^{2},&n\mbox{ even}.\end{array}\right\\}$

This is the sequence associated to the balanced quivers and exactly matches
our arithmetic in Subsection 3.2. In particular, the sharp points of the
infinite stairs associated to the toric diagram are given by the consecutive
quotients for $\\{A_{n}\\}$, which tend to $2+\sqrt{3}$. $\Box$

## References

  * [1] Mohammad Akhtar, Tom Coates, Sergey Galkin, and Alexander M. Kasprzyk.  Minkowski polynomials and mutations.  SIGMA Symmetry Integrability Geom. Methods Appl., 8:Paper 094, 17, 2012.
  * [2] Mohammad E. Akhtar and Alexander M. Kasprzyk.  Mutations of fake weighted projective planes.  Proc. Edinb. Math. Soc. (2), 59(2):271–285, 2016.
  * [3] V. I. Arnol’d.  Mathematical methods of classical mechanics, volume 60 of Graduate Texts in Mathematics.  Springer-Verlag, New York, [1989].  Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein, Corrected reprint of the second (1989) edition.
  * [4] Matthew Strom Borman, Tian-Jun Li, and Weiwei Wu.  Spherical Lagrangians via ball packings and symplectic cutting.  Selecta Math. (N.S.), 20(1):261–283, 2014.
  * [5] Keon Choi, Daniel Cristofaro-Gardiner, David Frenkel, Michael Hutchings, and Vinicius Gripp Barros Ramos.  Symplectic embeddings into four-dimensional concave toric domains.  J. Topol., 7(4):1054–1076, 2014.
  * [6] David A. Cox, John B. Little, and Henry K. Schenck.  Toric varieties, volume 124 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, 2011.
  * [7] Dan Cristofaro-Gardiner.  Symplectic embeddings from concave toric domains into convex ones.  J. Differential Geom., 112(2):199–232, 2019.
  * [8] Dan Cristofaro-Gardiner.  Ehrhart polynomials and symplectic embeddings of ellipsoids.  Journal Lon. Math. Soc., 112, 2020.
  * [9] Dan Cristofaro-Gardiner, Tara S. Holm, Alessia Mandini, and Ana Rita Pires.  Infinite staircases and reflexive polygons.  arXiv, 2020.
  * [10] B. Feng, Y.-H. He, K. D. Kennaway, and C. Vafa.  Dimer models from mirror symmetry and quivering amoebae.  Adv. Theor. Math. Phys., 12(3):489–545, 2008.
  * [11] Vladimir V. Fock and Alexander B. Goncharov.  Cluster ensembles, quantization and the dilogarithm.  Ann. Sci. Éc. Norm. Supér. (4), 42(6):865–930, 2009.
  * [12] Sergey Fomin and Andrei Zelevinsky.  Cluster algebras. I. Foundations.  J. Amer. Math. Soc., 15(2):497–529, 2002.
  * [13] J. Forsgård.  On dimer models and coamoebas.  Ann. Inst. Henri Poincaré D, 6(2):199–219, 2019.
  * [14] David Frenkel and Dorothee Müller.  Symplectic embeddings of 4-dim ellipsoids into cubes.  J. Symplectic Geom., 13(4):765–847, 2015.
  * [15] William Fulton.  Introduction to toric varieties, volume 131 of Annals of Mathematics Studies.  Princeton University Press, Princeton, NJ, 1993.  The William H. Roever Lectures in Geometry.
  * [16] Marco Golla and Laura Starkston.  The symplectic isotopy problem for rational cuspidal curves.  arXiv, 2020.
  * [17] A. L. Gorodentsev and A. N. Rudakov.  Exceptional vector bundles on projective spaces.  Duke Math. J., 54(1):115–130, 1987.
  * [18] M. Gromov.  Pseudo holomorphic curves in symplectic manifolds.  Invent. Math., 82(2):307–347, 1985.
  * [19] Mark Gross.  Tropical geometry and mirror symmetry, volume 114 of CBMS Regional Conference Series in Mathematics.  Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2011.
  * [20] Daniel R. Gulotta.  Properly ordered dimers, $R$-charges, and an efficient inverse algorithm.  J. High Energy Phys., (10):014, 31, 2008.
  * [21] Paul Hacking and Yuri Prokhorov.  Smoothable del pezzo surfaces with quotient singularities.  Compos. Math., 146(1):169–192, 2010.
  * [22] Jeff Hicks.  Tropical Lagrangians and Homological Mirror Symmetry.  arXiv:1904.06005, 2019.
  * [23] Richard Hind.  Lagrangian unknottedness in Stein surfaces.  Asian J. Math., 16(1):1–36, 2012.
  * [24] Michael Hutchings.  Embedded contact homology and its applications.  In Proceedings of the International Congress of Mathematicians. Volume II, pages 1022–1041. Hindustan Book Agency, New Delhi, 2010.
  * [25] Michael Hutchings.  Quantitative embedded contact homology.  J. Differential Geom., 88(2):231–266, 2011.
  * [26] Michael Hutchings.  Beyond ECH capacities.  Geom. Topol., 20(2):1085–1126, 2016.
  * [27] Nathan Owen Ilten.  Deformations of smooth toric surfaces.  Manuscripta Math., 134(1-2):123–137, 2011.
  * [28] Nathan Owen Ilten and Robert Vollmert.  Deformations of rational $T$-varieties.  J. Algebraic Geom., 21(3):531–562, 2012.
  * [29] Alexander Kasprzyk, Benjamin Nill, and Thomas Prince.  Minimality and mutation-equivalence of polygons.  Forum Math. Sigma, 5:e18, 48, 2017.
  * [30] P. W. Kasteleyn.  Graph theory and crystal physics.  In Graph Theory and Theoretical Physics, pages 43–110. Academic Press, London, 1967.
  * [31] J. Kollár and N. I. Shepherd-Barron.  Threefolds and deformations of surface singularities.  Invent. Math., 91(2):299–338, 1988.
  * [32] Francois Lalonde and Dusa McDuff.  The classification of ruled symplectic $4$-manifolds.  Math. Res. Lett., 3(6):769–778, 1996.
  * [33] Weonmo Lee, Yong-Geun. Oh, and Renato Vianna.  Asymptotic behavior of exotic Lagrangian tori $T_{a,b,c}$ in $\mathbb{CP}^{2}$ as $a+b+c\to\infty$.  arXiv:1904.11775, 2019.
  * [34] Naichung Conan Leung and Margaret Symington.  Almost toric symplectic four-manifolds.  J. Symplectic Geom., 8(2):143–187, 2010.
  * [35] Jun Li, Tian-Jun Li, and Weiwei Wu.  The symplectic mapping class group of $\mathbb{C}P^{2}\\#n\overline{\mathbb{C}P^{2}}$ with $n\leq 4$.  Michigan Math. J., 64(2):319–333, 2015.
  * [36] Tian-Jun Li and Weiwei Wu.  Lagrangian spheres, symplectic surfaces and the symplectic mapping class group.  Geom. Topol., 16(2):1121–1169, 2012.
  * [37] Cheuk Yu Mak and Helge Ruddat.  Tropically constructed Lagrangians in mirror quintic threefolds.  arXiv:1904.11780, 2019.
  * [38] Diego Matessi.  Lagrangian pairs of pants.  arXiv:1802.02993, 2018.
  * [39] Dusa McDuff.  Rational and ruled symplectic $4$-manifolds.  In Geometry of low-dimensional manifolds, 2 (Durham, 1989), volume 151 of London Math. Soc. Lecture Note Ser., pages 7–14. Cambridge Univ. Press, Cambridge, 1990.
  * [40] Dusa McDuff.  The structure of rational and ruled symplectic $4$-manifolds.  J. Amer. Math. Soc., 3(3):679–712, 1990.
  * [41] Dusa McDuff.  Symplectic embeddings of 4-dimensional ellipsoids.  J. Topol., 2(1):1–22, 2009.
  * [42] Dusa McDuff and Felix Schlenk.  The embedding capacity of 4-dimensional symplectic ellipsoids.  Ann. of Math. (2), 175(3):1191–1282, 2012.
  * [43] Grigory Mikhalkin.  Amoebas of algebraic varieties and tropical geometry.  In Different faces of geometry, volume 3 of Int. Math. Ser. (N. Y.), pages 257–300. Kluwer/Plenum, New York, 2004.
  * [44] Grigory Mikhalkin.  Decomposition into pairs-of-pants for complex algebraic hypersurfaces.  Topology, 43(5):1035–1065, 2004.
  * [45] Grigory Mikhalkin.  Enumerative tropical algebraic geometry in $\mathbb{R}^{2}$.  J. Amer. Math. Soc., 18(2):313–377, 2005.
  * [46] Grigory Mikhalkin.  Tropical geometry and its applications.  In International Congress of Mathematicians. Vol. II, pages 827–852. Eur. Math. Soc., Zürich, 2006.
  * [47] Grigory Mikhalkin.  Examples of tropical-to-Lagrangian correspondence.  Eur. J. Math., 5(3):1033–1066, 2019.
  * [48] Grigory Mikhalkin.  Spines for amoebas of rational curves.  arXiv:1906.04500, 2019.
  * [49] Alexei N. Rudakov.  Markov numbers and exceptional bundles on $P^{2}$.  Izv. Akad. Nauk SSSR Ser. Mat., 52(1):100–112, 240, 1988.
  * [50] Alexei N. Rudakov.  Exceptional vector bundles on a del pezzo surface.  In Algebraic geometry and its applications (Yaroslavl , 1992), Aspects Math., E25, pages 177–182. Friedr. Vieweg, Braunschweig, 1994.
  * [51] Felix Schlenk.  Symplectic embedding problems, old and new.  Bull. Amer. Math. Soc. (N.S.), 55(2):139–182, 2018.
  * [52] Margaret Symington.  Four dimensions from two in symplectic topology.  In Topology and geometry of manifolds (Athens, GA, 2001), volume 71 of Proc. Sympos. Pure Math., pages 153–208. Amer. Math. Soc., Providence, RI, 2003.
  * [53] Michael Usher.  Infinite staircases in the symplectic embedding problem for four-dimensional ellipsoids into polydisks.  Algebr. Geom. Topol., 19(4):1935–2022, 2019.
  * [54] R. Vianna.  Infinitely many exotic monotone Lagrangian tori in $\mathbb{CP}^{2}$.  J. Topol., 9(2):535–551, 2016.
  * [55] R. Vianna.  Infinitely many monotone Lagrangian tori in del Pezzo surfaces.  Selecta Math. (N.S.), 23(3):1955–1996, 2017.
  * [56] Renato Vianna.  Infinitely many exotic monotone Lagrangian tori in $\mathbb{CP}^{2}$.  J. Topol., 9(2):535–551, 2016.
  * [57] Renato Vianna.  Infinitely many monotone Lagrangian tori in del Pezzo surfaces.  Selecta Math. (N.S.), 23(3):1955–1996, 2017.