# Exploring the data of blockchain-based metaverses

S. Casale Brunet1,2, M. Mattavelli1, L. Chiariglione3 École Polytechnique
Fédérale de Lausanne, Switzerland
WhaleAnalytica.com, Switzerland
CEDEO SRL, Italy

###### Abstract

In recent years the concept of metaverse has evolved in the attempt of
defining richer immersive and interactive environments supporting various
types of virtual experiences and interactions among users. This has led to the
emergence of various different metaverse platforms that utilize blockchain
technology and non-fungible tokens (NFTs) to establish ownership of metaverse
elements and attach features and information to it. This article will delve
into the heterogeneity of the data involved in these metaverse platforms, as
well as highlight some dynamics and features of them. Moreover, the paper
introduces a metaverse analysis tool developed by the authors, which leverages
machine learning techniques to collect and analyze daily data, including
blockchain transactions, platform-specific metadata, and social media trends.
Experimental results are reported are presented with a use-case scenario
focused on the trading of digital parcels, commonly referred to as metaverse
real estate.

###### Index Terms:

metaverse, blockchain, NFT, machine learning

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##  I Introduction

The metaverse can be defined as a digital platform consisting of virtual
environments and virtual worlds enabled by various technologies. These virtual
environments can be created and customized by individuals or organizations for
a variety of purposes, including entertainment, education, communication, and
business. The metaverse can consist of multiple virtual layers, which can be
connected through metachains and secured through the use of blockchain
technology [1, 2, 3]. Its implementation may also require the use of
technologies such as virtual reality, augmented reality, and artificial
intelligence, depending on the specific use case [4], with the human
experience and interaction remaining a key component [5].

This paper focuses on a specific type of metaverse platform based on the
concept of virtual land parcels, in which blockchain technology is used to
enable ownership and representation of these digital assets. In other words,
these platforms implement the concept of real estate in a distributed,
trustless, and interactive environment. A comprehensive understanding of this
new digital asset class requires knowledge of topics such as traditional real
estate and financial markets, blockchain technology, cryptocurrency assets,
and non-fungible tokens (NFTs). Studies on blockchain-based metaverse
platforms, such as Decentraland and The Sandbox Game, have shown that the
location of virtual parcels is a key factor in determining their value, and
that the market for digital land represented as NFTs is similar to the market
for physical real estate [6, 7, 8].

This paper presents a technical analysis of the key components of blockchain-
based metaverse platforms based on virtual land parcels. It illustrates and
demonstrates how various data points, such as blockchain transactions, the
number of users connected to the platform, and social media engagement, can be
collected and effectively used to create accurate statistical models for
determining the value of each individual parcel within each metaverse. In
contrast to the state of the art, where studies focus on a specific platform
and generally only consider transactions on the blockchain, this study
presents a cross-sectional analysis of the top five Ethereum-based metaverses
in which all collected heterogeneous data is analyzed on a daily basis, giving
users the ability to assess the economic value of the parcels in these
platforms.

The paper is structured as follows: Section II provides an overview of the
main technical components of blockchain-based metaverse platforms based on
virtual land parcels and NFTs; Section III illustrates the different types of
data that can be extracted and collected from these platforms; Section IV
analyzes the collected data and demonstrates how it can be used to build
effective statistical models based on machine learning techniques to estimate
the fair economic value of each parcel; Finally, Section V concludes the paper
and discusses future research directions.

##  II Blockchain-based metaverse enviroments

Digital real estate markets within metaverses, also known as virtual worlds,
often exhibit characteristics similar to traditional real estate markets, such
as limited availability of land and the inability to easily move or transfer
ownership of property [6]. However, these markets utilize decentralized
technologies, such as blockchain and smart contracts, to facilitate secure and
trustless transactions. This means that individuals can directly participate
in the economy and own digital real estate, referred to as digital parcels,
without the need for a central authority to verify or mediate the transaction.

In this article, we examined five Ethereum-based platforms based on their
popularity, trading volume and our own expertise. These are: Voxels,
Decentraland, The Sandbox Game, Somnium Space and Otherside. It is important
to point out that the contents of this list are derived entirely from the
knowledge and expertise of the authors. It is important to note that this list
should not be interpreted as providing any form of financial advice, as it is
intended for informational purposes only.

##  III Data collection

To fully understand and evaluate the value of virtual worlds in metaverse
platforms, it is important to consider the types of data that can be analyzed
for each environment. These data can be classified into two categories: on-
chain data, which refers to financial transactions involving NFTs of parcels
and are stored on the blockchain, and off-chain data such as parcel
descriptions (e.g., location, size) and utilization (e.g., traffic patterns)
that are generally not persistent and available from centralized servers.
These data must be aggregated and carefully organized in order to be analyzed
effectively. In the following sections, we will explore the main types of data
that make up these metaverses and how we have implemented these data in our
daily data acquisition and analysis tool, shown in Figure 1. We have made this
tool publicly accessible [9] and we have also developed an API that allows
users to retrieve heterogeneous data from various metaverses with a common
semantics, ensuring that the data is always up-to-date. In the following, we
present a discussion on the various types of data.

Figure 1: Metaverse analysis frameworks developed in the paper.

a) Metaverse-specific data: information about each parcel in a virtual world,
including its location and maximum build size, is often stored on centralized
servers and represented as JSON files. This information, known as metadata, is
usually encoded according to the de-facto ERC721 metadata standard or the
Enjin metadata recommendations. More advanced information about a parcel can
typically be obtained through the metaverse platform’s public API (e.g., see
[10] for Decentraland, and [11] for Voxels). b) Blockchain transactions: the
data stored on public blockchains, which are publicly accessible by design,
can be efficiently accessed when structured in formats such as SQL [12]. The
metaverse environments in this study are primarily based on Ethereum (with
some secondary use of Polygon). The data collection techniques used are the
ones we described in [13]. c) NFT exchange-specific data: parcels can be
traded on exchanges specific to the platform, such as the Decentraland
Marketplace, and on more broadly deployed exchanges like OpenSea, LooksRare,
and others. The information about the parcels that are on sale, including
their price, is not stored on the blockchain but rather on the exchange
website. To keep track of this information, it may be necessary to use the
exchange API (if available, e.g., OpenSea API [14]) or web scraping
techniques. Two interesting insights to consider when looking at this list of
parcels for sale are the lowest price on the list, also known as the floor
price, and the size of the list. The size of the list, along with the number
of daily transactions, can give an indication of how ’liquid’ the collection
trading is. d) Media and social media popularity The popularity and social
community of cryptocurrencies and NFT assets in mainstream and social media is
a very important factor. In fact, studies such as [15] have emphasized this
phenomenon. It is therefore important to monitor the sentiment on main social
media platforms (e.g., Twitter, Reddit, Google). This can provide insight into
the popularity of each metaverse platform and the broader concept of metaverse
which, as we will see in the next sections, are correlated on the average
price of the parcels.

##  IV Data Analysis and Visualisation

In the following, we describe how we analyzed the market for the five
metaverses described in Section II for the period from January 1, 2021 to
November 30, 2022. First, we will describe the techniques implemented for data
collection, followed by the types of analysis carried out: starting with a
global market analysis, then for each separate platform, and ending with the
implementation of a machine learning model where, using the available data, it
has been possible to define a suitable value for each land in the various
metaverses.

###  IV-A Dataset

We obtained information on the blockchain transactions of the parcel NFTs and
the average daily price of cryptocurrencies related to each metaverse (e.g.,
SAND and MANA) using the Dune platform [16]. This platform provides SQL-
structured and querable data on major blockchains, including raw transactions
data and aggregated views (e.g., NFT trades). Using the official Twitter API,
we collected data on social trends by gathering all tweets mentioning the
accounts of each project and those containing the ”#metaverse” term hashtag,
as well as the Google trend for the ”metaverse” term. For each metaverse
platform, specific information was then gathered based on the information
available from their metadata. This is summarized in Table I. All of the
resulting data and metadata we obtained were saved in our local database, as
illustrated in Figure 1 where the metaverse analysis framework we developed is
shown. Table II summarizes the volumes of trades in USD and the number of
tweets. For the purpose of this study, we considered only transactions with an
economic value (i.e., not those where only the owner of the token associated
with the parcel has changed without a corresponding economic exchange). We
also filtered these transactions by eliminating, for each project, those above
the 99th percentile. According to the table, the total volume of transactions
that we considered was approximately USD 1,500M and included approximately
160k transactions (with 10% of the total volume and 1% of the transactions
already subtracted). At this stage, we did not perform any filtering on the
collected tweets.

TABLE I: Collected and analysed features for each metaverse platform.

| The Sandbox Game | Decentraland | Voxels | Somnium Space | Otherside
---|---|---|---|---|---
Blockchain transactions |  | \- Daily, weekly, monthly volume in ETH, USD, and metaverse specific currencies
---
\- Daily, weekly, monthly number of transactions
\- Daily, weekly, monthly parcel average price
\- Daily ETH/USD and metaverse specific currencies spot price
Exchange information |  | \- Daily listed parcels
---
\- Daily listings price distribution
Parcel metadata |  | \- X, Y coordinates
---
\- Proximity to POIs
\- Estate size
| \- X, Y coordinates
---
\- Proximity to POIs
\- Estate size
| \- X, Y coordinates
---
\- Proximity to POIs
\- Volume
| \- X, Y coordinates
---
\- Area
| \- X, Y coordinates
---
\- Sediment and Artifact
\- Koda
Parcel traffic |  | \- Hourly max. users | \- Daily cumulative users | \- Hourly max. users |
Social media popularity |  | \- Number of daily tweets for topic and discussion related to the specific platform and broader metaverse topic
---
\- Google popularity related to the metaverse topic

TABLE II: Dataset size considering the time period from 1st January 2021 to
30th November 2022 included. Statistics on discarded data refer to the value
of the project.

Metavers | Volume (USD) | Transactions | Tweets
---|---|---|---
Considered | Discarded | Considered | Discarded
Otherside | 906,704,668 | 59.14% | 138,340,545 | 13.24% | 67,563 | 42.12% | 683 | 1.00% | 1,015,177 | 1.73%
Voxels | 33,393,758 | 2.18% | 3,378,292 | 9.19% | 5,565 | 3.47% | 58 | 1.03% | 493,085 | 0.84%
Decentraland | 112,433,708 | 7.33% | 8,290,684 | 6.87% | 11,218 | 6.99% | 114 | 1.01% | 4,151,326 | 7.07%
Somnium Space | 22,311,548 | 1.46% | 2,231,310 | 9.09% | 2,210 | 1.38% | 23 | 1.03% | 129,348 | 0.22%
The Sandbox Game | 458,337,419 | 29.89% | 27,184,674 | 5.60% | 73,844 | 46.04% | 746 | 1.00% | 5,046,455 | 8.59%
Total | 1,533,181,103 |  | 179,425,508 | 10.48% | 160,400 |  | 1,624 | 1.00% | 58,740,001 |

###  IV-B Metaverse market trends

During the period we studied, several notable events occurred: Facebook
rebranded itself to Meta in late October 2021, leading to a surge in
mainstream interest in the term ”metaverse”; the rapid growth of the
cryptocurrency market, driven primarily by Ethereum and Bitcoin [17], reached
all-time highs in November 2021; the Otherside platform was launched on 30th
April, 2022; successively the market as a whole saw contraction and crash in
both equities and cryptocurrency due to challenging macroeconomic conditions.
These events likely had an impact on the trend in digital land sales for the
five metaverses we analyzed, as shown in Figure 2.

###  IV-C Platform-specific market trends

We can further delve into which metaverse platform had the most success in
terms of trade volume and social media engagement by examining Figure 2. We
can see that all collections saw the number of transactions and their average
value increase following the explosion of interest in the metaverse topic in
November 2021, and then followed the downward trend that began in spring 2022
(Figures 2(b) and 2(a)). The overall market considering all the five projects
might not have been negatively impacted, however this is only due to the
launch of ”Otherside” by the creators of the BAYCs (which is one of the most
successful and influential collection in the NFT market today). In fact,
”Otherside” has managed to become one of the metaverse projects with the most
traded volume in a short period of time (see Table II). It is interesting to
see the distribution of daily transactions versus average daily price
illustrated in Figure 2(c): from here, we can see that the market is clustered
into two main groups, with ”The Sandbox Game” and ”Otherside” forming one
group and the remaining collections forming the other. By analyzing the
exchanges where these transactions take place, we estimated that approximately
88% of the USD transaction volume occurs on OpenSea, while the next two most-
used exchanges are x2y2 and the Decentraland (DCL) marketplace (note that in
this latter only Decentraland parcels can be traded) with approximately 6% and
3.6%, respectively. We also find that ETH and WETH are the most common crypto-
currencies used for trading, accounting for 80% and 10% of the total USD
volume, respectively. WETH, which are ERC-20 tokens that represent 1:1 ETH,
are often used to purchase parcels (and other NFTs) through a bidding process.
Bids are usually placed below the current lowest price of the collection,
known as the floor price. Once a parcel has been acquired, it may be resold in
an attempt to make a (quick) profit. This is known as flipping. During times
when the market is experiencing a negative trend, such as a liquidation phase,
there may be an increase in the number of accepted bids for WETH. This can be
seen in Figure 2(d), which shows the ratio (represented by the green line)
between the daily trading volume of WETH and other currencies. This ratio
tends to increase significantly when the market is experiencing a negative
trend and average parcel prices are declining.

(a) Parcels daily trades average price

(b) Number of parcel daily trades

(c) Parcels number of daily trades and average price distribution

(d) Parcels monthly volume per exchange. The green line, right y-axis,
represents the ratio between WETH and other cryptocurrencies.

Figure 2: Parcel trading volume and number of transactions.

###  IV-D Parcels position, geometry and traffic

In the previous section, we analyzed various metaverses individually,
examining the average daily price of parcels sold. If we instead focus on
individual parcels, recent studies have shown that location is a key factor
that can significantly impact the parcel value when compared to the average
value. For example, studies [6] and [8] on Decentraland and The Sandbox Game
respectively have both concluded that, despite the absence of travel distance
in the metaverse, location is extremely important. These studies, however,
focus on two specific platforms where the size of each parcel is uniform. In
the more general case of Voxels and Somnium Space parcel size may also affect
the price of a parcel. Therefore, the framework we implemented (shown in
Figure 1) also gathers the metadata for each parcel, including information
about the available area and volume for construction on the parcel. In
addition, for Decentraland, Somnium Space, and Voxels, we have also collected
information about the traffic on each parcel. In the following, we analyze the
information we have collected for each individual parcel in addition to their
geographical location, as shown in Table I. a) Voxels: each parcel has
different dimensions, with associated height and area limits for building. For
each parcel, we are able to obtain the daily cumulative number of unique users
who have logged in. b) Decentraland: all the parcels have the same size of 16m
x 16m, but adjacent parcels can be grouped into estates. As of now, there are
approximately 2,160 estates. For each parcel, we are able to collect the
number of concurrent users connected per hour: Figure 3 shows the maximum
number of users connected to the platform from June 2022 to the end of
November 2022 (the period for which we have data). c) The Sandbox Game: all
the parcels have the same size of 96m x 96m, but adjacent parcels can be
grouped into estates in fixed sizes of 3x3, 6x6, 12x12, and 24x24 parcels. d)
Somnium Space: there are three types of parcels with different sizes: ’S’
(2,000m3), ’M’ (15,000m3), and ’XL’ (75,000m3). For each plot, we collect the
number of connected users per hour, distinguishing between spectators and
players. d) Otherside: for each parcel, we identify sediments, artifacts, and
the possible presence of one of the 10,000 Koda NFTs.

Figure 3: Decentraland traffic traffic defined as maximum users connected per
parcel per hour.

###  IV-E Machine learning models of metaverses

To examine potential correlations and build a statistical model to determine
the economic value of each parcel, we first collected and organized data from
various sources and at different levels. We then used a Spearman correlation
analysis to analyze the following variables: daily average price, volume, and
number of parcel sales; metaverse topic popularity on Google (measured through
Google Trends); daily tweets related to the specific metaverse platform and
the metaverse topic in general; and the daily dollar price of ETH and any
platform’s cryptocurrency. The results are displayed in Figure 4. In order to
improve the accuracy of the analysis, we first removed any seasonal components
from each time series. We can see that the average price and trading volume
are strongly correlated with the number of tweets for the Otherside platform
(see Figure 4(b)), while for the other projects, there seems to be a stronger
link with popularity on other channels, as indicated by the correlation with
Google Trends (e.g., see Figure 4(a)). This probably indicates that Twitter is
less influential for NFT metaverse projects compared to what was observed for
example in [15] for NFT profile pictures (PFP) projects. We believe that the
current nature of Otherside’s trading and its underdeveloped gaming
environment make it more akin to a PFP project rather than a metaverse one.
The second step was to understand in more detail which variables most
influence the selling price of a plot. To do this, we used XGBoost (eXtreme
Gradient Boosting) [18], a widely-used machine learning algorithm that is
particularly effective for regression and classification tasks. We conducted
separate experiments for each platform, training the model to predict the
prices of the plots based on the other available data. We randomly divided the
dataset described in Table II into two parts: a training set containing 80% of
the transactions for each platform, and a test set containing the remaining
20%. We then evaluated the model’s accuracy and reliability using the test set
by comparing its predictions to the actual sale prices of each plot
transaction (e.g., see Figure 5(a)). A randomized search for hyperparameter
tuning was used to identify the best parameters configuration for each model.
The number of features and accuracy of each metaverse model are summarized in
Table III. In general, we found that parcel location (in terms of x, y
coordinates) is the factor that most influences the sale price on each
metaverse, as already demonstrated in [6, 7, 8]. However, we can add that
other factors with a significant influence on the selling price of a parcel
are the average daily price of other plots sold, the daily price of ETH (which
can also serve as a general crypto market indicator), and the level of
activity on a parcel, as for example in the case of Decentraland (see Figure
5(b)). The results of this study indicate that user traffic on a parcel is not
a significant determinant of its price. Instead, factors related to the
revenue-generating potential of the parcel are more likely to play a role. In
our opinion, this is because we are currently in an exploratory phase of the
market, where individuals and organizations investing in digital parcels are
primarily focused on acquiring strategic locations as a form of marketing
investment.

(a) Decentraland

(b) Otherside

Figure 4: Spearman correlations for some of the collected data. TABLE III:
Parcel price estimation model size and accuracy.

Metaverse | Model Size | Accuracy %
---|---|---
Number of features | Training | Performance
The Sandbox Game | 8 | 95.8 | 86.0
Decentraland | 11 | 91.3 | 88.7
Voxels | 21 | 98.0 | 75.1
Somnium Space | 12 | 95.1 | 88.3
Otherside | 11 | 87.5 | 71.5

(a) Estimated price (x-axis) v.s. actual sale price (y-axis)

(b) Features importance

Figure 5: Decentraland parcel price estimation model.

###  IV-F Data exploration and visualization tool

We have created an exploration tool to help users navigate and explore data
more easily. This tool is available at [9] and allows users to browse
different platforms at various levels, displaying various types of information
directly on the plots. These include: 1) Land view, which colors parcels based
on their characteristics in the metaverse (e.g., size); 2) Trading view, which
highlights parcels for sale on different exchanges with different colors based
on their sale price; 3) Last price view, which colors parcels based on the
last sale price; 4) Value view, which colors parcels based on the ratio
between the sale price and our estimated price; 5) Fair value view, which
colors parcels based on our estimated fair value price; 6) Flip view, which
uses color to indicate how many times the parcel has been traded. Depending on
the structure of a particular metaverse, there may also be specific metrics
such as: 7) Traffic view, which uses color to highlight the most heavily
trafficked parcels; and 8) Resources view, which uses color to indicate the
availability of different resources.

##  V Conclusions

In this article, we conducted a technical analysis of five major blockchain-
based metaverse platforms that implement the concept of digital land and real
estate in a decentralized digital environment. We described the various
technological components and how various types of data, such as blockchain
transactions, parcel traffic, and engagement on social networks, can be
effectively extracted and analyzed from these platforms. The results obtained
have been: 1) the development of the first cross-platform metaverse data
collection and analysis tool, data that is now accessible through a public and
unified API; 2) the systematic creation of machine learning models that,
through data fusion and curation, are able to estimate a fair value of each
individual parcel in each metaverse; 3) the verification, thanks to these
models, that location is a generally fundamental factor in determining the
value of a parcel, as already demonstrated bysome state-of-the-art work. In
comparison to these studies, which only focus on two specific platforms, our
work has been performed on the five main Ethereum-based platforms. Future
studies will aim to improve the accuracy of these estimation models and study
more complex traffic patterns, for example by testing whether it is possible
to distinguish between real users and bots.

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# A Markov theorem for plat closure of surface braids in Dunwoody and periodic
Takahashi manifolds

Alessia Cattabriga  Alma Mater Studiorum, Bologna, Italy  Paolo Cavicchioli
Alma Mater Studiorum, Bologna, Italy

(February 2022)

###### Abstract

In this article we deal with the problem of finding equivalence moves for
links in Dunwoody and periodic Takahashi manifolds. We represent these
manifolds using Heegaard splitting and we represent the embedded links as plat
closure of elements in the braid group of the corresponding Heegaard surfaces.
More precisely, starting from an open Heegaard diagram for such manifolds, we
determine the plat slide equivalence moves algorithmically and compute them
explicitly in some cases.

Keywords: plat closure of braids, links in 3-manifolds, Dunwoody manifolds,
Takahashi Manifolds, algorithm.

MSC Classification: 57K10, 57K30, 57-08, 57-04

##  1 Introduction

The representation of links as plat closures of braids with an even number of
strands goes back to the works of Hilden [18] and Birman [4], who proved that
any link in $\mathbb{R}^{3}$ may be represented as the plat closure of a
braid. Since then, much work has been done using this representation in the
direction of studying the equivalence problem of links, for example by
defining and analyzing link invariants as the Jones polynomial (see [3, 5]).
Using Heegaard splittings, in [13] Doll introduced the notion of
$(g,b)$-decomposition or generalized bridge decomposition for links in a
closed, connected and orientable (from now on c.c.o.) $3$-manifold, opening
the way to the study of links in 3-manifolds via surface braid groups and
their plat closures [2, 7, 25]. The equivalence of links in 3-manifolds, under
isotopy, has recently been described in [6], where the authors find a finite
set of moves that connect braids in $B_{g,2n}$, the braid group on $2n$
strands of a surface of genus $g$, having isotopic plat closures. In their
result some moves are explicitly described as elements in $B_{g,2n}$, while
others, called plat slide moves, depend on a Heegaard surface for the manifold
and are explicitly described only for the case of Heegaard genus one, that is
for lens spaces and $S^{2}\times S^{1}$. Moreover, in [11] an algorithm to
determine plat slide moves for manifolds with Heegaard genus two is presented.
In this paper we deal with the same problem, but in the case of two infinite
families of manifolds: Dunwoody and (periodic) Takahashi manifolds.

Dunwoody manifolds are c.c.o. 3-manifolds introduced in [14] by means of a
graph with a cyclic symmetry determining an open Heegaard diagram. Some
interesting results connect these manifolds and cyclic branched covering of a
$(1,1)$-knot (i.e., knots admitting a $(1,1)$ decomposition): in [8] and [15]
it has been proven that the class of Dunwoody manifolds coincides with the
class of strongly-cyclic branched coverings of $(1,1)$-knots. In [9] the
family of Dunwoody manifold has been generalized to include also manifolds
with non-empty boundary, while, more recently, other graph with cyclic
symmetry generalizing those introduced in [14] have been studied (see [19]).

Takahashi manifolds are c.c.o. 3-manifolds introduced in [24] by Dehn surgery
with rational coefficients along a specific $2n$-component link in $S^{3}$
(see Figure 8). Several important classes of 3-manifolds, such as (fractional)
Fibonacci manifolds [17, 26] and Sieradski manifolds [10], represent notable
examples of periodic Takahashi manifolds. These have been intensively studied
in many papers as [20, 22, 23, 26]. Among Takahashi manifolds the periodic
ones are those whose surgery coefficients present an order $n$ cyclic
symmetry: in [22] the author proves that these manifolds are cyclic coverings
of the connected sum of two lens spaces or $S^{3}$ branched over a
$(2,1)$-knot.

In the paper, we determine algorithmically plat slide moves for links in these
manifolds: in order to do so, we use the fact that both families admit a
symmetric open Heegaard diagram depending of a finite number of integer
parameters. Our approach could be easily generalized to other families of
manifolds with the same property.

The paper is organized as follows. In Section 2 we recall the definition of
Heegaard diagram, plat closure of links in 3-manifolds and the equivalence
moves introduced in [6]. In Section 3 (resp. 4), after giving the definition
of the family of Dunwoody (resp. Takahashi) manifolds, we compute the
equivalence moves for the considered family of 3-manifolds, together with some
notable examples.

In the following, manifolds are always assumed to be closed, connected and
orientable and are considered up to homemorphisms. Links inside them, that is,
closed 1-dimensional smooth submanifolds, will be considered up to isotopy.

##  2 Preliminaries

In this section we briefly recall the notion of Heegaard splitting and
Heegaard diagram for 3-manifolds as well as the definition of plat closure of
links in 3-manifolds.

###  2.1 Heegaard splittings

Let $M$ be a 3-manifold, a _Heegaard spitting_ for $M$ is the data of a triple
$(H^{*},H,\phi)$, where $H^{*}$ and $H$ are two copies of a genus $g$ oriented
handlebody (see Figure 1) standardly embedded in $\mathbb{R}^{3}$ and
$\phi:\partial H\rightarrow\partial H^{*}$ is an orientation reversing
homeomorphism such that $M=H\cup_{\phi}H^{*}$. The surface $\partial
H\cup_{\phi}\partial H^{*}\subseteq M$ has genus $g$ and is called _Heegaard
surface_ for $M$.

Figure 1: Standard genus $g$ handlebody and generators of the surface braid
group.

Each $3$-manifold admits Heegaard splittings [16]: we call _Heegaard genus_ of
a 3-manifold $M$ the minimal genus of a Heegaard surface for $M$.
For instance, the $3$-sphere $S^{3}$ is the only $3$-manifold with Heegaard
genus $0$, while the manifolds with Heegaard genus $1$ are lens spaces (i.e.,
cyclic quotients of $S^{3}$) and $S^{2}\times S^{1}$. While $S^{3}$, as well
as lens spaces, have, up to isotopy, only one Heegaard surface of minimal
genus (and those of higher genera are stabilizations of that of minimal
genus), in general, a manifold may admit non isotopic Heegaard surfaces of the
same genus (see [21] for the case of Seifert manifolds).
Each Heegaard splitting $(H^{*},H,\phi)$ can be represented by means of a
(closed) _Heegaard diagram_ that is a triple
$(\Sigma,\mathbf{d}^{*},\mathbf{d})$, with $\Sigma=\partial H^{*}$ and
$\mathbf{d}^{*}=\\{d_{1}^{*},\dots,d_{g}^{*}\\}$ (resp.
$\mathbf{d}=\\{d_{1},\dots,d_{g}\\}$) a set of meridians111We recall that a
_set of meridians_ for a handlebody of genus $g$ is a collection of closed
curves $\mathbf{d}^{*}=\\{d^{*}_{1},\dots,d^{*}_{g}\\}$ on its boundary, such
that the curves bound properly embedded pairwise disjoint disks
$D_{1},\dots,D_{g}$ (dotted in Figure 1) and cutting the handlebody along
these disks yields a 3-ball. for $H^{*}$ (resp. the image in $\Sigma$, via
$\phi$, of a set of meridians for $H$). Moreover, one of the two system of
meridians, say $\mathbf{d}^{*}$ could be always choose in a standard way, as
in Figure 1. If we cut $\Sigma$ along $\mathbf{d}^{*}$, we obtain a sphere
with $2g$ holes, say $D_{1},\dots,D_{2g}$, distinct and paired, each pair
corresponding to a certain meridian $d_{i}^{*}$. Moreover the curves of
$\mathbf{d}$ will be naturally cut into arcs joining the holes in various
ways, giving a graph on the sphere with $2g$ holes called _open Heegaard
diagram_ of $(H^{*},H,\phi)$ and still denoted by
$(\Sigma,\mathbf{d}^{*},\mathbf{d})$.
On the contrary, if we have an open Heegaard diagram, in order to obtain the
closed one we just need to identify the $D_{i}$’s. It can be done by attaching
1-handles connecting the paired disks and arcs along the handles, connecting
the paired vertices of the graph.
The result will be a set of closed curves which will represent the system
$\mathbf{d}$.

###  2.2 Plat closure of links in 3-manifolds

Let $M$ be a 3-manifold, fix a Heegaard diagram
$(\Sigma,\mathbf{d}^{*},\mathbf{d})$ for $M$ so that $\Sigma$ is the boundary
of a standard handlebody and $\mathbf{d}^{*}$ are chosen in a standard way
(see Figure 1); denote with $g$ the genus of $\Sigma$. Consider
$B_{g,2n}=\pi_{1}(C_{2n}(\Sigma),*)$ the braid group on $2n$ strands of
$\Sigma$, that is, the fundamental group of the configuration space of $2n$
(unordered) points $P_{1},\ldots,P_{2n}$ in $\Sigma$. A presentation of
$B_{g,2n}$ is given in [1]. Referring to Figure 1 the generators are
$\sigma_{1},\ldots,\sigma_{2n-1}$, the standard braid ones, and
$\alpha_{1},\ldots,\alpha_{g},\beta_{1},\ldots,\beta_{g}$, where $\alpha_{i}$
(resp. $\beta_{i}$) is the braid whose strands are all trivial except the
first one which goes once along the $i$-th longitude (resp. $i$-th meridian)
of $\Sigma$. Moreover, we fix a set of $n$ disjoint arcs
$\rho_{1},\ldots,\rho_{n}$ embedded into $\Sigma$, such that
$\partial\rho_{i}=\\{P_{2i-1},P_{2i}\\}$, for $i=1,\ldots,n$, as in Figure 1.
Now we can associate to each element $\gamma\in B_{g,2n}$ a link in $M$,
called the _plat closure_ of $\gamma$, and denoted with $\hat{\gamma}$,
obtained “closing” a geometric representative of $\gamma$ in
$\Sigma\times[0,1]$ by connecting $P_{2i-1}\times\\{0\\}$ with
$P_{2i}\times\\{0\\}$ through $\rho_{i}\times\\{0\\}$ and
$P_{2i-1}\times\\{1\\}$ with $P_{2i}\times\\{1\\}$ through
$\rho_{i}\times\\{1\\}$, for $i=1,\ldots,n$.

For each link $L$ in $M$, there exists a braid
$\gamma\in\cup_{n\in\mathbb{N}}B_{g,2n}$ such that $\hat{\gamma}=L$. Moreover,
as proved in [6, Theorem 3], two braids
$\gamma_{1},\gamma_{2}\in\bigcup_{n\in\mathbb{N}}B_{g,2n}$ have isotopic plat
closures if and only if $\gamma_{1}$ and $\gamma_{2}$ differ by a finite
sequence of the following moves

$\displaystyle(M1)$
$\displaystyle\qquad\sigma_{1}\gamma\longleftrightarrow\gamma\longleftrightarrow\gamma\sigma_{1}$
$\displaystyle(M2)$
$\displaystyle\qquad\sigma_{2i}\sigma_{2i+1}\sigma_{2i-1}\sigma_{2i}\gamma\longleftrightarrow\gamma\longleftrightarrow\gamma\sigma_{2i}\sigma_{2i+1}\sigma_{2i-1}\sigma_{2i}$
$\displaystyle(M3)$
$\displaystyle\qquad\sigma_{2}\sigma_{1}^{2}\sigma_{2}\gamma\longleftrightarrow\gamma\longleftrightarrow\gamma\sigma_{2}\sigma_{1}^{2}\sigma_{2}$
$\displaystyle(M4)$
$\displaystyle\qquad\alpha_{j}\sigma_{1}^{-1}\alpha_{j}\sigma_{1}^{-1}\gamma\longleftrightarrow\gamma\longleftrightarrow\gamma\alpha_{j}\sigma_{1}^{-1}\alpha_{j}\sigma_{1}^{-1}\quad\textup{for
}j=1,\ldots,g$ $\displaystyle(M5)$
$\displaystyle\qquad\beta_{j}\sigma_{1}^{-1}\beta_{j}\sigma_{1}^{-1}\gamma\longleftrightarrow\gamma\longleftrightarrow\gamma\beta_{j}\sigma_{1}^{-1}\beta_{j}\sigma_{1}^{-1}\quad\textup{for
}j=1,\ldots,g$ $\displaystyle(M6)$
$\displaystyle\qquad\gamma\longleftrightarrow T_{k}(\gamma)\sigma_{2k}$
$\displaystyle\ \ \quad\textup{ where }T_{k}:B_{g,2n}\to B_{g,2n+2}\textup{ is
defined by }T_{k}(\alpha_{i})=\alpha_{i},\ T_{k}(\beta_{i})=\beta_{i}\textup{
and }$ $\displaystyle\ \ \quad
T_{k}(\sigma_{i})=\left\\{\begin{array}[]{l}\sigma_{i}\qquad\qquad\qquad\qquad\qquad\qquad\quad\textup{if
}i<2k\\\
\sigma_{2k}\sigma_{2k+1}\sigma_{2k+2}\sigma_{2k+1}^{-1}\sigma_{2k}^{-1}\
\qquad\quad\textup{if }i=2k\\\
\sigma_{i+2}\qquad\qquad\qquad\qquad\qquad\qquad\textup{if }i>2k\\\
\end{array}\right.$ $\displaystyle(psl_{i}^{*})$
$\displaystyle\qquad\gamma\longleftrightarrow\gamma\beta_{i}\quad\textup{for
}i=1,\dots,g$ $\displaystyle(psl_{i})$
$\displaystyle\qquad\gamma\longleftrightarrow\bar{d}_{i}\gamma,\quad\textup{for
}i=1,\dots,g,$

where $\bar{d}_{i}\in B_{g,2}$ is a braid representative of $d_{i}$ with $2$
strands.222Since $d_{i}$ bounds a disk in $M$ it is always possible such a
representation. So in order to determine completely the moves, it is necessary
to describe the $\bar{d}_{i}$ as a word in terms of the braid generators. From
now on we will call psl-moves the moves $psl_{i}$.

###  2.3 From Heegaard diagrams to psl-moves

In [6] an explicit formula for psl-moves in 3-manifold with Heegaard genus one
is computed, while in [11], an algorithm to compute the psl-moves for
3-manifolds of Heegaard genus two is described, starting from an open Heegaard
diagram of the manifold. However, the same idea works in higher genus, so we
briefly recall it.

Let $(\Sigma,\mathbf{d}^{*},\mathbf{d})$ be an open Heegaard diagram for $M$.
In the corresponding closed one on $\Sigma$, we color in blue the arcs along
the 1-handles arising from the identifications of the paired disks, and in red
the other arcs. Note that, by construction, moving along each $d_{i}$, any red
arc is followed by a blue arc and vice-versa.

In order to represent $d_{i}$’s curves in terms of generators of the braid
group we can proceed as follows: (1) orient each curve and divide it into
elementary pieces of couples of subsequent red-blue arcs, (2) connect the two
ends of each piece with a fixed base point by fixed arcs, so to obtain an
elementary closed curve, (3) catalogue each elementary closed curve arising in
this way up to homotopy and (4) represent it in terms of generators of the
surface braid group. This procedure gives rise to a dictionary that can be
used to implement a computer algorithm that takes as input the graph
corresponding to the open Heegaard diagram and gives as output the words
$\bar{d}_{i}$’s corresponding to the psl-moves. It just works by visiting the
cycles of the graph and replacing each elementary arc with the corresponding
word in the dictionary.

##  3 Dunwoody manifolds

Let $a,b,c,n\in\mathbb{N}$, with $n>0$ and $a+b+c>0$. Let
$\Gamma=\Gamma(a,b,c,n)$ be the regular, trivalent, planar graph depicted in
Figure 2. The graph contains $n$ top and $n$ bottom circles, each having
$d=2a+b+c$ vertices; we denote them with $D_{1}^{u},\dots,D_{n}^{u}$ and
$D_{1}^{d},\dots,D_{n}^{d}$ respectively (all the indexes are considered mod
$n$). For each $i=1,\dots,n$, there are $a$ parallel arcs, called upper (resp.
lower) arcs, connecting $D_{i-1}^{u}$ (resp. $D_{i-1}^{d}$) with $D_{i}^{u}$
(resp. $D_{i}^{d}$), $b$ parallel arcs, called diagonal arcs, connecting
$D_{i}^{u}$ to $D_{i-1}^{d}$ $c$ parallel arcs, called vertical arcs,
connecting $D_{i}^{u}$ to $D_{i}^{d}$. Clearly, $\Gamma$ has a cyclic symmetry
of order $n$ sending $D_{i}^{u}$ into $D_{i-1}^{u}$ and $D_{i}^{d}$ into
$D_{i-1}^{d}$, for $i=1,\ldots,n$.

Figure 2: The graph $\Gamma(a,b,c,n)$ and the labelling of its vertices. An
arc with label $k$ identifies $k$ parallel arcs.

The one-point compactification of the plane brings $\Gamma$ onto $S^{2}$. Now,
let $r,s$ two given integers: we give a clockwise (resp. counterclockwise)
orientation to each $D_{i}^{u}$ (resp. $D_{i}^{d}$) and we enumerate the
vertices as in Figure 2 (all the numbers are to be considered mod $d$), for
$i=1,\dots,n$. If we cut of from $S^{2}$ the region of the $2n$ disks bounded
by each $D_{i}^{u}$ and $D_{i}^{d}$ and not containing any arc of $\Gamma$,
and we glue $D_{i}^{u}$ with $D_{i+s}^{d}$, for $i=1,\ldots,n$, so that
vertices having the same labelling correspond, we obtain a regular graph of
degree four in an orientable surface $\Sigma_{n}$ of genus $n$. Of course, by
construction, we can always consider $r$ mod $d$ and $s$ mod $n$.

Through the identification, the $nd$ arcs of $\Gamma$ connect with each other
via their endpoints creating $m$ closed curves $e_{1},\dots,e_{m}$, where
$e_{i}$ is the curve passing from the vertex labelled $a+b+1$ in $D_{i}^{u}$.
Now, denote with $d_{i}=D_{i}^{u}=D_{i+s}^{d}$ and set
$\mathcal{D}=\\{d_{1},\dots,d_{n}\\},\mathcal{E}=\\{e_{1},\dots,e_{m}\\}$.

Naturally, if we cut $\Sigma_{n}$ along $\mathcal{D}$ we do not disconnect the
surface; if $m=n$ and even cutting $\Sigma_{n}$ along $\mathcal{E}$ does not
disconnect the surface, then $(\Sigma_{n},\mathcal{D},\mathcal{E})$ is a
Heegaard diagram of genus $n$ of a 3-manifold, completely determined by the
6-tuple $(a,b,c,n,r,s)$: we call such a manifold _Dunwoody manifold_.

Clearly not all the 6-tuples $\sigma=(a,b,c,n,s,r)\in\mathbb{Z}^{6}$ determine
a Dunwoody manifold, that is, are such that the set $\mathcal{E}$ contains
exactly $n$ closed curves not disconnecting the surface $\Sigma_{n}$. We call
admissible such 6-tuples and with $D(\sigma)$ (resp. $H(\sigma)$), the
Dunwoody manifold (resp. open Heegaard diagram) associated to $\sigma\in S$.

In Figure 3 we depict the case of $M(1,1,1,3,2,1)$ that is homeomorphic to
$S^{1}\times S^{1}\times S^{1}$ (see [8]).

Figure 3: The Dunwoody manifold $M(1,1,1,3,2,1)$ homeomorphic to $S^{1}\times
S^{1}\times S^{1}$: the three curves of the $\mathcal{E}$ system are depicted
with different colors and different (or no) labels.

###  3.1 The dictionary for Dunwoody manifolds

Given a Dunwoody manifold $M=M(a,b,c,n,s,r)$, we use the open Heegaard diagram
of Figure 2 for the construction of the dictionary. The elementary blue-red
pieces are those corresponding to the four type of arcs of the graph, i.e.,
upper, lower, diagonal and vertical, each with two possible orientation. We
use $P_{1}$ as base point (see Figure 1) and, in the figures, we color in grey
the fixed arcs connecting the elementary pieces with the base point.

  * •

Horizontal arc, upper or lower (see Figure 4): we denote with
$\overrightarrow{A^{U}_{i}}$ (resp. $\overrightarrow{A^{L}_{i}}$) an upper
(resp. lower) arc connecting $D_{i}^{u}$ (resp. $D_{i}^{d}$) with
$D_{i-1}^{u}$ (resp. $D_{i-i}^{d}$), and denote with
$\overleftarrow{A^{U}_{i}}$ (resp. $\overleftarrow{A^{L}_{i}}$) an upper
(resp. lower) arc connecting $D_{i}^{u}$ (resp. $D_{i}^{d}$) with
$D_{i+1}^{u}$ (resp. $D_{i+i}^{d}$)

$\displaystyle\overrightarrow{A^{U}_{i}}$
$\displaystyle=\beta_{i-1}^{-1}\alpha_{i-1}$
$\displaystyle\overleftarrow{A^{U}_{i}}$ $\displaystyle=\beta_{i}\alpha_{i+1}$
$\displaystyle\overrightarrow{A^{L}_{i}}$ $\displaystyle=\alpha_{i-s-1}^{-1}$
$\displaystyle\overleftarrow{A^{L}_{i}}$ $\displaystyle=\alpha_{i-s+1}^{-1}.$

  * •

Vertical arc (see Figure 5): the description of the elementary piece
corresponding to the vertical arc depends on the gluing parameter $s$ of
6-tuple determining $M$. We denote with $C_{i}^{s}$ the elementary blue-red
piece corresponding to a vertical arc connecting the disks $D_{i}^{u}$ and
$D^{d}_{i}$ with parameter $s$ (so that $D^{u}_{i}$ is glued to
$D^{d}_{i+s}$), while we use the arrow on the left to denote the orientation

$\displaystyle\downarrow\\!C_{i}^{s}$
$\displaystyle=w_{i,s}\alpha_{i+n-s}^{-1}$ $\displaystyle\uparrow\\!C_{i}^{s}$
$\displaystyle=w_{i,s}^{-1}\alpha_{i}$

where

$w_{i,s}=\prod_{j=0}^{k}\beta_{i+j},\quad 1\leq k\leq n\textup{ and }k\equiv
n-s-1\textup{ mod }n.$

  * •

Diagonal arc: from the construction of the Dunwoody manifold it is quite
straightforward to see that the elementary curve corresponding to a diagonal
arc with gluing parameter $s$ is equal to the one corresponding to a vertical
arc with gluing parameter $s+1$, that is

$\downarrow\\!B_{i}^{s}=\downarrow\\!C_{i}^{s+1}$
$\uparrow\\!B_{i}^{s}=\uparrow\\!C_{i+1}^{s+1},$

where $\downarrow\\!B_{i}^{s}$ denotes the elementary piece corresponding to a
diagonal arc connecting the disks $D_{i}^{u}$ and $D^{d}_{i-1}$ with parameter
$s$, and $\uparrow\\!B_{i}^{s}$ denotes the elementary piece corresponding to
a diagonal arc connecting the disks $D_{i}^{d}$ and $D^{u}_{i+1}$ with
parameter $s$.

Figure 4: The four possible horizontal arcs.

Figure 5: An example of elementary curve corresponding to a vertical arc - in
this case $\downarrow\\!C_{3}^{3}=\beta_{1}\beta_{3}\alpha_{2}^{-1}$.

######  Example 3.1.

In the case of the Dunwoody manifold $M(1,1,1,3,2,1)$, homeomorphic to
$S^{1}\times S^{1}\times S^{1}$ (see [8]) and depicted in Figure 3, if we
denote with $e_{1}$ the blue curve (label $\times$), $e_{2}$ the green one
(label $\circ$) and $e_{3}$ the red one (no label) we have, starting from the
vertex labeled 3:

$\begin{array}[]{ll}\overline{e_{1}}&=\beta_{1}\beta_{2}\alpha_{3}^{-1}\beta_{3}\alpha_{1}\beta_{3}^{-1}\alpha_{3}\alpha_{1}^{-1}\\\
\overline{e_{2}}&=\beta_{2}\beta_{3}\alpha_{1}^{-1}\beta_{1}\alpha_{2}\beta_{1}^{-1}\alpha_{1}\alpha_{2}^{-1}\\\
\overline{e_{3}}&=\beta_{3}\beta_{1}\alpha_{2}^{-1}\beta_{2}\alpha_{3}\beta_{2}^{-1}\alpha_{2}\alpha_{3}^{-1}.\end{array}$

###  3.2 Fibonacci and Sieradski manifolds

Recall that the Minkus manifold $M_{n}(2a+1,2r)$ is the $n$-fold cyclic
covering branched on the 2-bridge knot $b(2a+1,2r)$, with $(2a+1,2r)=1$. In
[15] it is proved that $M_{n}(2a+1,2r)=M(a,0,1,n,r,\bar{s})$, where $\bar{s}$
can be computed on $\Gamma(a,0,1,n,r,0)$ as follows: consider the vertex
labelled $v=a+b+1$ of $D^{u}_{1}$, and orient downwards the arc connecting
this vertex with $D^{d}_{1}$ and call $e_{1}$ the cycle of $\mathcal{E}$
passing from $v$. Now, follow along this orientation the arcs of $\Gamma$
belonging to $e_{1}$, and count the arcs running from $D^{u}_{i-1}$ to
$D^{u}_{i}$ or $D^{d}_{i-1}$ to $D^{d}_{i}$ and those running from $D^{u}_{i}$
to $D^{u}_{i-1}$ or $D^{d}_{i}$ to $D^{d}_{i-1}$. The difference between the
first type of arcs and the second one is $\bar{s}$.

For $r=1$ and $a=2$, the knot $b(5,2)$ is the figure-eight knot, so
$M(2,0,1,n,1,\bar{s})$ is a Fibonacci manifold, while when $r=a=1$, the knot
$b(3,2)$ is the trefoil knot, so $M(1,0,1,n,1,\bar{s})$ is a Sieradski
manifold. For both families of manifolds it is possible to compute explicitly
the words related to the curves:

  * •

Fibonacci case $M(2,0,1,n,1,\bar{s})$: we have $\bar{s}=0$ fal all
$n\in\mathbb{N}$; so the curve $e_{i}$ starting in the vertex labeled 3 of
$D_{i}^{u}$ is the red curve (and thicker curve) in Figure 6, with related
word:

$\overline{e_{i}}=\alpha_{i}^{-1}\beta_{i-1}^{-1}\alpha_{i-1}\alpha_{i}^{-1}\beta_{i}\alpha_{i+1}\alpha_{i}^{-1}.$

  * •

Sieradski case $M(1,0,1,n,1,\bar{s})$: we have $\bar{s}=-2$ for $n>2$; so the
curve $e_{i}$ starting in the vertex labeled 2 of $D_{i}^{u}$ is the red curve
(and thicker curve) in Figure 7, with related word:

$\overline{e_{i}}=\beta_{i}\beta_{i+1}\alpha_{i+2}^{-1}\beta_{i+1}^{-1}\alpha_{i+1}\alpha_{i}^{-1}.$

Figure 6: The general case of a Fibonacci manifold, described by the 6-tuple
$M(2,0,1,n,1,0)$ Figure 7: The general case of a Sieradski manifold, described
by the 6-tuple $M(1,0,1,n,1,-2),n>2$

##  4 Periodic Takahashi manifolds

Let ${\cal L}_{n}\subseteq S^{3}$ be the link with $2n$ components represented
in Figure 8: it is a closed chain of $2n$ unknotted components having a cyclic
symmetry of order $n$ which permutes the unknotted components.

The Takahashi manifold
$T_{n}(p_{1}/q_{1},\ldots,p_{n}/q_{n};r_{1}/s_{1},\ldots,r_{n}/s_{n})$ is the
manifold obtained by Dehn surgery on $S^{3}$, along the link ${\cal L}_{n}$,
with surgery coefficients alternatively equal to $p_{k}/q_{k}$ and
$r_{k}/s_{k}$, with $1\leq k\leq n$. Without loss of generality, we can always
suppose $\gcd(p_{k},q_{k})=1$, $\gcd(r_{k},s_{k})=1$ and $p_{k},r_{k}\geq 0$.
If $p_{1}/q_{1}=p_{2}/q_{2}=\cdots=p_{m}/q_{m}=p/q$ and
$r_{1}/s_{1}=r_{2}/q_{2}=\cdots=r_{m}/s_{m}=r/s$, i.e., the surgery
coefficients have the same cyclic symmetry of order $n$ of the link ${\cal
L}_{n}$, the Takahashi manifold is called periodic and it is denoted simply by
$T_{n}(p/q,r/s)$.

Figure 8: Presentation via Dehn surgery for the Takahashi manifold
$T_{n}(p_{1}/q_{1},\ldots,p_{n}/q_{n};r_{1}/s_{1},\ldots,r_{n}/s_{n})$.

In [12, Proposition 7] the authors describe an open Heegaard diagram for a
periodic Takahashi manifold $T_{n}(p/q,r/s)$ in the case $p/q,r/s\geq 0$ and
$p/q,r/s\neq 1$, that, up to isotopy, is the one depicted in Figure 9, where
the couples of corresponding circles are $D_{i}^{u},D_{i}^{d}$ for
$i=1,\dots,2n$ glued according to the orientation and so that the fat red
points are identified; without loss of generality we always assume
$p,q,r,s\geq 0$. In Figure 13 we depict the case of $T_{2}(1/2,2/3)$.

(a) $p>q,r>s$

(b) $p<q,r<s$

(c) $p>q,r<s$

(d) $p<q,r>s$

Figure 9: Four open Heegaard diagrams describing, depending on the parameters
$p,q,r,s$, the periodic Takahashi manifold. An arc with label $k$ identifies
$k$ parallel arcs.

###  4.1 The dictionary for periodic Takahashi manifolds

For the construction of the dictionary in the setting of Takahashi manifolds,
we refer to the Heegaard diagram of Figure 9: we have eight type of elementary
arcs and we denote them according to Figure 10. Following Figure 11 down
below, we can describe the words associated to the elementary red-blue arcs.
As in the Dunwoody case, we use $P_{1}$ as the base point and, in the figures,
we color in grey the arcs connecting the elementary pieces with the base
point.

  * •

Arcs $A^{U}$ and $A^{L}$: they are as the horizontal arcs in the Dunwoody case
so the result is the same (see Figure 4)

$\displaystyle\overrightarrow{A^{U}_{2i}}$
$\displaystyle=\beta_{2i+1}^{-1}\alpha_{2i+1}$
$\displaystyle\overleftarrow{A^{U}_{2i}}$
$\displaystyle=\beta_{2i+1}\alpha_{2i}$
$\displaystyle\overrightarrow{A^{L}_{2i}}$ $\displaystyle=\alpha_{2i+1}^{-1}$
$\displaystyle\overleftarrow{A^{L}_{2i}}$ $\displaystyle=\alpha_{2i}^{-1}.$

  * •

Arcs $B$ and $C$: these coincide with vertical or diagonal arcs of the
Dunwoody case with $s=0$, so we have

$\displaystyle\overrightarrow{B_{2i-1}}$ $\displaystyle=\alpha_{2i}^{-1}$
$\displaystyle\overleftarrow{B_{2i-1}}$ $\displaystyle=\alpha_{2i-1}$
$\displaystyle\downarrow\\!C_{2i-1}$ $\displaystyle=\alpha_{2i-1}^{-1}$
$\displaystyle\uparrow\\!C_{2i-1}$ $\displaystyle=\alpha_{2i-1}.$

  * •

Arc $F$: we obtain the curve depicted in Figure 11. In order to find the
corresponding word we need to decompose the original loop into three elements,
see Figure 12. Following these three elementary pieces we obtain

$\displaystyle\overrightarrow{F_{2i}}$
$\displaystyle=[\alpha_{2i+1},\beta_{2i+1}]\beta_{2i+2}^{-1}\alpha_{2i+2}=\alpha_{2i+1}^{-1}\beta_{2i+1}^{-1}\alpha_{2i+1}\beta_{2i+1}\beta_{2i+2}^{-1}\alpha_{2i+2}$
$\displaystyle\overleftarrow{F_{2i}}$
$\displaystyle=\beta_{2i}[\beta_{2i-1},\alpha_{2i-1}]\alpha_{2i-2}=\beta_{2i}\beta_{2i-1}^{-1}\alpha_{2i-1}^{-1}\beta_{2i-1}\alpha_{2i-1}\alpha_{2i-2}$

where $[\alpha,\beta]=\alpha^{-1}\beta^{-1}\alpha\beta$.

  * •

Arcs $G$, $X$ and $Y$: for all these kind of arcs the procedure is similar to
the previuos one, obtaining the following sequence of moves

$\displaystyle\overrightarrow{G_{2i-1}}$
$\displaystyle=\beta_{2i}^{-1}\beta_{2i+1}^{-1}\alpha_{2i+1}$
$\displaystyle\overleftarrow{G_{2i-1}}$
$\displaystyle=\beta_{2i-1}\beta_{2i-2}\alpha_{2i-3}$
$\displaystyle\overrightarrow{X_{2i-1}}$
$\displaystyle=\beta_{2i+1}^{-1}\alpha_{2i+2}^{-1}$
$\displaystyle\overleftarrow{X_{2i-1}}$
$\displaystyle=\beta_{2i-1}^{-1}\alpha_{2i-2}^{-1}$
$\displaystyle\overrightarrow{Y_{2i-1}}$ $\displaystyle=\alpha_{2i+1}^{-1}$
$\displaystyle\overleftarrow{Y_{2i-1}}$ $\displaystyle=\alpha_{2i-3}^{-1}.$

Figure 10: All the possible elementary arcs in the periodic Takahashi case.

Figure 11: From top to bottom, the curves corresponding to $B$-arcs, $C$-arcs,
$D$-arcs, $E$-arcs, $F$-arcs and $G$-arcs. Figure 12: The case of an arc of
type $\overrightarrow{F}$. Figure 13: The Takahashi manifold $T_{2}(1/2,2/3)$.

######  Example 4.1.

In the case of the Takahashi manifold $T_{2}(1/2,2/3)$ depicted in Figure 13,
if we denote with $e_{1}$ the blue curve (label $\times$), $e_{2}$ the red one
(no label), $e_{3}$ the green one (label $\circ$) and $e_{4}$ the black one
(label $\star$), following the arrow we have

$\begin{array}[]{lll}\overline{e_{1}}&=\overrightarrow{B_{1}}\overleftarrow{F_{2}}\overrightarrow{X_{3}}\overleftarrow{F_{2}}\overrightarrow{A^{L}_{4}}&=\alpha_{2}^{-1}\beta_{2}\beta_{1}^{-1}\alpha_{1}^{-1}\beta_{1}\alpha_{1}\alpha_{4}\beta_{1}^{-1}\alpha_{2}^{-1}\beta_{2}\beta_{1}^{-1}\alpha_{1}^{-1}\beta_{1}\alpha_{1}\alpha_{4}\alpha_{1}^{-1}\\\
\overline{e_{2}}&=\overrightarrow{B_{3}}\overleftarrow{F_{4}}\overrightarrow{X_{1}}\overleftarrow{F_{4}}\overrightarrow{A^{L}_{2}}&=\alpha_{4}^{-1}\beta_{4}\beta_{3}^{-1}\alpha_{3}^{-1}\beta_{3}\alpha_{3}\alpha_{2}\beta_{3}^{-1}\alpha_{4}^{-1}\beta_{4}\beta_{3}^{-1}\alpha_{3}^{-1}\beta_{3}\alpha_{3}\alpha_{2}\alpha_{3}^{-1}\\\
\overline{e_{3}}&=\overrightarrow{A^{U}_{2}}\overleftarrow{Y_{3}}\overrightarrow{G_{1}}\overleftarrow{Y_{3}}\overrightarrow{B_{1}}\overrightarrow{A^{U}_{2}}\overleftarrow{Y_{3}}\overrightarrow{B_{1}}&=\beta_{3}^{-1}\alpha_{3}\alpha_{1}^{-1}\beta_{2}^{-1}\beta_{3}^{-1}\alpha_{3}\alpha_{1}^{-1}\alpha_{2}^{-1}\beta_{3}^{-1}\alpha_{3}\alpha_{1}^{-1}\alpha_{2}^{-1}\\\
\overline{e_{4}}&=\overrightarrow{A^{U}_{4}}\overleftarrow{Y_{1}}\overrightarrow{G_{3}}\overleftarrow{Y_{1}}\overrightarrow{B_{3}}\overrightarrow{A^{U}_{4}}\overleftarrow{Y_{1}}\overrightarrow{B_{3}}&=\beta_{1}^{-1}\alpha_{1}\alpha_{3}^{-1}\beta_{4}^{-1}\beta_{1}^{-1}\alpha_{1}\alpha_{3}^{-1}\alpha_{4}^{-1}\beta_{1}^{-1}\alpha_{1}\alpha_{3}^{-1}\alpha_{4}^{-1}.\end{array}$

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