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What Couples Are Saying
Lauren and Gregg's backyard wedding in Grosse Ile, Michigan
Posted By : Carl Bower/
Lauren and Gregg are two of the most amazing people I've had the pleasure to work with as a Denver wedding photojournalist. I had flown out to Washington DC to do a corporate shoot and was just packing up my equipment in the building lobby when they approached me and asked if I could make some portraits for their company website. They had been watching me work and thought I would be a good fit. When I sent them the pictures, they called me and asked me to shoot their wedding. They've been quite successful navigating the complexities of federal and state government real estate, but when it comes to major decisions they tend to follow their instincts.
Lauren with the officiant, soon to be her brother in law.
Lauren's father walks her down the aisle.
These two had chemistry in spades. They are a unit, they are a team, and two of the kindest people I've ever met.
Everyone seems to offer advice on where and how to sign a marriage license…
…but no one tells you not to flip the pages over the votive candles. Their signatures were blacked out, but I think the contract holds.
Some people think you have to have two photographers to cover all angles at an event, but photojournalists manage to do it all the time. The trick is to anticipate, and to move. It works pretty well.
Gregg's daughter with his mother. He had a few kids going into his relationship with Lauren, and this was a pivotal day in their lives as well. When kids are involved, it's not just the union of a couple, but of a family. There are complex emotions present, and I feel they are very much a part of the story.
Gregg's dad after his toast…
…and Lauren's sisters during their roast.
Gregg with his youngest son.
Not everyone is a natural on the dance floor.
Family matters. It's their day too.
When Gregg's mom talks to someone, she is so warm and present they feel they are the only other person on earth. I wanted to sneak her onto the plane back to Denver.
Little stories are everywhere. As a Denver wedding photojournalist, I love weddings because I can be in the midst of all these dynamics and no one ever asks why I'm there. I'm the help. I'm like the guy refilling water glasses, in plain sight but also invisible. It's liberating.
Gregg and Lauren have built a retreat and started a foundation, Restore the Restorers, which will someday become a movement to provide restoration to first responders, counselors, therapists, and veterans, people who devote their lives to helping others. They take care of everyone else, but who takes care of them? Their jobs are critical, but the burnout rate is high as they become mentally and emotionally depleted. These two now devote most of their time and resources to those so easily overlooked. To spend any time at all with them is to find this not only unsurprising, but inevitable.
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Nicole & Chad's History Center Colorado wedding
What I do outside of documentary wedding photography
Michelle and Nick's Denver wedding at Coohills restaurant
Why I'm often the last to leave.
Photojournalist and Pulitzer finalist specializing in authentic wedding photography. Based in Denver, Colorado and serving clients nationwide.
Portraits and Projects
In addition to weddings, I continue to focus on long-term documentary projects and commissioned portraits for magazines and individuals. You can see my other work at carlbower.com.
Copyright © Carl Bower. All Rights Reserved.


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Ecuador: Chinese company ordered to stop mining in ground-breaking legal judgment
madaraka.online June 6, 2018 Uncategorized
Source: mining.com (2018-06-03)
A Chinese company has been ordered by an Ecuadorian judge to halt a mining project which, he says, violates indigenous peoples rights over their own land and natural resources.
The organisation which brought the case claims that this sets a precedent in the country, because it'is "the first time that the justice system sides with Indigenous communities and not with the government or foreign companies".
Judge orders Chinese company to stop mining activities in Ecuadorian town
A judge from the Ecuadorian province of Cuenca ruled that Ecuagoldmining South America S.A, owned by Chinese investment firm Junefield Group S.A., violated article 57 of the country's Constitution, which establishes people's right to be consulted prior to the development of any resource exploitation project in their communities.
Judge Paúl Serrano said that given such violation, Ecuagoldmining should immediately halt operations at its Río Blanco gold/silver project. His decision was in response to a motion presented by Ecuarunari, an organization that represents 12 ethnics groups that belong to the Kichwa nation, whose members alleged that there were legal gaps in the consultation process for Río Blanco and that, therefore, the environmental permits granted were not valid.
Ecuarunari also argued that the company was polluting water sources, which is considered very dangerous given that the project is located in the area of influence of the Cajas National Park.
Project development started in the early 2000s but it was in 2016 when the Chinese miner began exploiting the 5,708-hectare site, located between the towns of Molleturo and Chaucha in the Cuenca municipality of the southern Azuay province. Production is estimated at 800 tonnes of ore per day.
According to El Comercio newspaper, activities at the mine were halted almost a month ago following protests by anti-mining groups.
In parallel with the rallies, Ecuarunari presented the aforementioned motion before Azuay's Court of Justice. The audience to hear their case ran for about four hours last Friday. Once all parties presented their arguments, the judge decided that it was urgent to stop mining operations at the site, to remove all military personnel from the area and to ask the Ombudsman's Office to make sure that the ruling is observed.
The decision can be appealed before the Provincial Court of Justice. However, Ecuagoldmining has not made public whether it is going to pursue a new legal action.
In a video published on Ecuarunari's Twitter page, the organization's president Yaku Pérez said that this decision sets a precedent because it is the first time that the justice system sides with Indigenous communities and not with the government or foreign companies.
Previous India: Bloody police killings of civil protestors against Vedanta's copper operations
Next Ghana: Newmont penalised for involvement in six mineworker deaths

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Feb. 23 – Investor Information Session in North Grenville
by David Mazur-Goulet | Jan 25, 2017 | Events
OREC's 600+ members are building a community-owned renewable energy future in Eastern Ontario; profitably, sustainably, and responsibly. Class A Preference Shares and Member Investment Notes are now available for purchase! These securities are available to members...
Building Local Economies through Cooperative Ownership
by David Mazur-Goulet | Mar 20, 2015 | Learn
It is common belief that companies build our economy. Most of us assume that strong businesses build a strong local economy. But most corporations operate on the premise that their prime responsibility is to maximize return for the shareholders while NOT breaking...
EmPowering Local Economies
by David Mazur-Goulet | Feb 12, 2014 | Learn
By Tim Inkpen Since the financial crisis of 2008, there has been a resurgence of interest in local economies. The crisis has caused many to look for ways of buffering their communities from drastic upheavals over which they little or no control. The basic concept...


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Tele2 Russia expands its GSM operations into 17 new regions
Press release Tele2 Group Nov 16 2007, 9:48 AM CET
Stockholm - Tele2 AB ("Tele2"), (Stockholm Stock Exchange: TEL2 A and TEL2 B), Europe's leading alternative telecom operator, today announced that it has been awarded three new GSM 1800 licenses in the regions of Novosibirsk, Tomsk and Evenkia in Siberia. The three regions have a total population of 3.7 million. Tele2 plan to invest in mobile networks and launch commercial service in these regions by the end of 2009.
In addition to the three awarded licenses the tender commission has allocated an additional 14 licenses to Tele2 and it is expected that the formal award of these licenses will occur shortly. The total 17 licenses cover a population of 19.1 million.
The allocation of new GSM 1800 licenses creates the opportunity for Tele2 to remain the fastest growing mobile operator in Russia. Tele2 will, in accordance with the license terms, launch mobile networks in the awarded regions end of 2009.
Further information can be obtained from:
Lars-Johan Jarnheimer, President and CEO Tele2 AB, Telephone: +46 8 5626 4000
Johnny Svedberg, Market Area Director Baltic & Russia, Telephone: +46 8 5626 4000
Lars Torstensson, Investor Inquiries, Telephone: +46 702 73 48 79
Lena Krauss, Investor Inquiries, Telephone: +46 8 5620 0045
Tele2 is Europe's leading alternative telecom operator Tele2's mission is to provide cheap and simple telecoms for everyone in Europe. Tele2 always strives to offer the market's best prices. We have 25 million customers in 17 countries. Tele2 offers fixed and mobile telephony, broadband, data network services and cable TV. Ever since Jan Stenbeck founded the company in 1993, it has been a tough challenger to the former government monopolies and other established providers. Tele2 has been listed on OMX Nordic Exchange since 1996. In 2006 we had operating revenue of SEK 50.3 billion and we reported an operating profit (EBITDA) of SEK 5.7 billion. Please visit us at www.tele2.com
Map Tele2 Russia


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Prism Place specializes in custom acrylic awards, trophies, plaques and medals. Each item is manufactured or assembled by a person with a disability, creating jobs and tax-paying citizens. No matter the occasion, our dedicated staff can help make your event shine. We offer standard designs you can tailor to your needs, or work with our talented team for a truly unique award.
Prism Place is pleased to quote your product designs that fall outside of our standard product line. We specialize in custom designs. Please contact us for pricing and lead times.
To get started with a custom award, contact us at 405-946-1079 x1600, email us at sales@prismplace.org or visit us at 2501 N. Utah Ave. in Oklahoma City during normal business hours.


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PCC Cleaning & Restoration is the Premier Choice for Damage & Restoration Providers in OKlahoma and NW Arkansas. We are locally owned and operated and have been in business since 2001 in Tulsa, OK. Because of our reputation in the industry, we are a respected restoration provider for many insurances companies, and are well respected by our clients.
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Storms left behind damage in Tulsa, OK Sunday morning, August 6, 2017 when an EF2 tornado touched down in the area after 1 a.m., according to the NWS. The NWS also confirmed three other tornadoes in the area. City officials say around 150 businesses were damaged and 12 were condemned.
PCC Cleaning & Restoration's Catastrophic Response Team was called in to action immediately to begin the mitigation phase of the project to help prevent puther damage to the business affected by the storm. After mitigating further damage, the clean-up and restoration phase began at several of the business affected by this local disaster.
PCC Cleaning & Restoration was proud to be trusted by several local businesses in the affected area to handle the complete project of returning their business to normal operations as quickly as they could. Please watch the video to the right to see some of the ensuing damage and ultimately the return to normal life after this storm.
PCC Cleaning and Restoration is a full service local restoration company. We start with quick response, then help you or your business throughout the entire process. From working with your insurance company, cleaning up the mess, or rebuilding the structure, we are here to help!
PCC Cleaning & Restoration has years of experience putting homes and businesses back to normal after all types of disasters and emergencies. Our clients count on us for quality and value, whether it's a single room, small to large house or small, medium or large business.


SUMMARY: After a successful launch on Friday, Russian space officials seem to have misplaced their Demonstrator-2 inflatable spacecraft. The spacecraft blasted from a Russian submarine into space on Friday atop a converted Volna SS-N-18 intercontinental ballistic missile. It then inflated to form a sail and returned to Earth somewhere near the Kamchatka Peninsula. Helicopters are still combining the area searching for remnants of the vehicle.


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London-Travel.info
Travel Guide for London
Tour Stonehendge
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December 19, 2018 No comments Sights
Tate Modern
Posted by LondonTravel
The Tate Modern is one of the top 5 museums in London. Admission is free!
It is one of the most important museums of modern art in the world. It was only opened in 2000 and is located in the centre of London on the south bank of the Thames. It is housed in a former power station with a huge turbine room, which was extensively rebuilt under the direction of the architects Herzog & de Meuron from Switzerland.
The most popular excursion by boat on the Thames
Meanwhile, a large part of London tourists take a boat trip on the Thames. Of course, everyone wants to see the center of London from the ship. Thus, the most popular boat tour is from Westminster to Greenwich, that is, across the city center of London. Very often booked tour, you should book online in advance:
>>> At Getyourguide.com you can buy tickets for the ships on the river Thames
The Tate Modern is officially called "Tate Gallery of Modern Art". However, this long name is rarely used. It is currently considered to be the most famous and largest museum of modern art in the world. The exhibition space was even significantly increased again for the 2012 Olympics in London.
In the Tate Modern one finds almost only works of art from the 20th century. Famous artists whose works are exhibited in the Tate Modern are Picasso, Beuys, Dali, Matisse and of course the superstar Andy Warhol.
Millenium Bridge: Most impressive is the "journey" on foot to the Tate Modern, over the Millenium Bridge (opened like the Tate Modern in 2000). It is an over 300 meter long pedestrian bridge across the Thames. The architect of the hypermodern bridge is none other than master architect Normen Forster himself. The Millenium Bridge connects the Tate Modern with the St Pauls Cathedral on the north side of the Thames. The bridge was also made famous by one of the Harry Potter films.
Our excursion tip: Harry Potter tour from London to the film studios outside the city. It is currently the most popular day trip from London >>> More information
Admission Tate Modern 2023: As in many major national museums in London, admission to the Tate Modern is free. This is one of the reasons why the Tate Modern is one of the most visited museums in the world with 4 million visitors per year. However, there are always special exhibitions that usually cost 10 pounds or more to enter. You can also get rid of money in the big restaurant on the top floor or in the big and good art bookstore in the building. Of course, there are also guided tours through the Tate Modern Art Museum.
Opening hours Tate Modern London 2023: Like most major museums in London, the Tate Modern is open seven days a week. The Tate Modern is open on Friday and Saturday from 10 a.m. to 2 p.m. and on other days from 10 a.m. to 6 p.m. The Tate Modern is only closed over Christmas.
How to get to the Tate Modern in London: Southwark Underground Station (Jubilee Line) is about 500 metres from the Tate Modern. If you want to walk over the Millenium Bridge (see above), the St Pauls (Central Line) or Mansion House (District Line and Circle Line) underground station is convenient. From both it is about 1 km to the Tate Modern, 300 meters of which are on the Millenium Bridge. One should, as everywhere in Central London, not get the idea to travel by car. There are no parking places around the Tate Modern. The city centre of London costs a toll.
Another great museum of Modern Art is the Guggenheim Museum Venice.
Our page about St Vitus Cathedral in Prague, the largest church in the city.
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http://phildobbie.com/wp-content/uploads/2018/11/121206twistedwire.mp3
AUDIO Twisted Wire
Is Google too big for its own bots?
By phildobbie	— 9 years ago
Google is serving up search results that give preference to its own commercial operations. That's the allegation from Brussels, Belgium-based lawyer David Wood, the legal counsel for the Initiative for a Competitive Online Marketplace (ICOMP).
In this week's Twisted Wire, he talks about how the search engine is now extending its influence to the mobile and mapping space. He suggests that the integrated ecosystem the company has developed is resulting in fewer clicks for competitors, creating an environment wherein innovative challengers to Google's emerging areas of influence are left out in the cold.
Has he got a point, or do we have a choice? If we are concerned, can't we simply move to another search provider? We could, but the point is, we don't. So Google's influence and control continues unabated.
It's a fascinating discussion about the emergence of an entrenched monopoly. And, of course, the more Google gathers data, the more sophisticated its offering is, and the harder it is for anyone to break the cycle. Unless the regulators step in.
What do you think? Is there a need for regulators to step in and control Google's dominance, to ensure that opportunities exist for competition in the space in which they operate?
AUDIO CrossTalk
Copyright law will help the Government's website blocking
By phildobbie	—	7 years ago
A government review has concluded that there are only minor changes needed to how agencies enforce website blocking, but it seems to ignore the impending changes to copyright law.
AUDIO BTalk
Even Floods Can't Dampen Optimism
By phildobbie	—	11 years ago
The cost of the Queensland floods could amount to as much as 3 percent of GDP. Despite this, CEOs around the country are still bullish about the future.
Thinking in Colour with Mind Maps
Tony Buzan, the inventor of MindMaps, explains why it is important to embrace a pictorial representations of your thoughts.


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\section{Introduction}

A key way to study closed symplectic manifolds is to break them down into two more easily understood parts: a neighborhood of a \emph{divisor} and a complementary \emph{Weinstein domain}. A divisor is a symplectic submanifold of co-dimension $2$. One can allow this submanifold to have certain controlled singularities, such as normal crossing singularities or more general singularities modeled on complex hypersurfaces. Donaldson proved that every symplectic manifold has a divisor \cite{Donaldson} and Giroux proved that this divisor can be chosen such that the complement of a regular neighborhood admits the structure of a Weinstein domain \cite{GirouxICM,Giroux}. A \emph{Weinstein domain} is a symplectic manifold with convex contact type boundary which can be broken down into symplectic handles modeled and glued as described by Weinstein \cite{Weinstein}. The symplectic topology of a Weinstein manifold is encoded in the attaching spheres of the handles. The attaching spheres can be represented by isotropic and Legendrian knots in the front projection. This Weinstein handlebody diagram gives a combinatorial/diagrammatic method to encode a symplectic manifold. There is a calculus of moves which relates different diagrams for equivalent Weinstein manifolds \cite{Gompf,DingGeiges}.

Recently, there has been increased study of symplectic divisors in symplectic manifolds, particularly in the case when the complement is Weinstein. Some of the motivation comes from homological mirror symmetry, where generalizing the link between coherent sheaves and Fukaya categories to larger classes of manifolds has required one to consider a mirror pair that includes not only a space, but also a divisor \cite{AurouxTduality}. One way to associate a Fukaya category for a divisor pair, is to look at the wrapped Fukaya category of the complement of the divisor. The Weinstein handle decomposition is key to understanding the wrapped Fukaya category, due to recent results that the co-cores of the handles generate the category \cite{CDGG, GPS1, GPS2}. The Floer homology of these co-cores is intrinsically tied to the Legendrian DGA of the Weinstein handlebody diagram~\cite{BEE,Ekholm, EkholmLekili} which is combinatorially calculated by Ekholm-Ng \cite{EkholmNg}.\smallskip

An important class of symplectic manifolds are \emph{toric manifolds}. These have been studied extensively as they form a large class of examples of integrable systems because of the symmetry provided by the Hamiltonian action of a torus on such manifolds. According to the famous Delzant classification, all compact symplectic toric $2n$-manifolds are uniquely (up to equivariant symplectomorphism) determined by convex $n$-dimensional polytopes, which correspond to the orbit space of the action. Much of the symplectic information can be encoded in the combinatorics of these polytopes known as Delzant polytopes.
 Moreover, toric manifolds have their origin in algebraic geometry, and they come by definition with a fibration by tori given by the action, so that they have been among the first cases of interest for homological mirror symmetry, especially in view of the SYZ philosophy (see for instance~\cite{Abouzaid}).
Every compact toric symplectic manifold is naturally equipped with a toric divisor. This is precisely the set of all points with non trivial stabilizer and the fixed points of the toric action are normal crossing singularities of the divisor. This can also be understood in terms of the moment map image polytope: the toric divisor is the preimage of the faces of the polytope under the moment map. The complement of a neighborhood of the divisor is symplectomorphic to a Weinstein domain whose completion is $T^*T^n$. (The complement of the divisor is the preimage of the interior of the polytope under the moment map, which is $T^n\times P$ where $P$ is a convex open subset of $\mathbb{R}^n$.) Hypersurfaces with normal crossing singularities can naturally be deformed to become less singular at the expense of increasing the topological complexity of the divisor and its complement. A toric manifold, together with its toric divisor or any smoothing of the divisor is a log Calabi-Yau pair which is a convenient setting for studying mirror symmetry of a space with a divisor~\cite{GrossHackingKeel, HackingKeating}.\smallskip

A manifold of dimension $4$ will have symplectic surface divisors. Normal crossing singularities in this dimension are just positive transverse intersections of two smooth branches, or \emph{nodes}. A deformation of this node smooths out the surface, trading the node for an annular tube which thus joins two different components or increases the genus of the surface. For a toric $4$-manifold, the complement of the (fully singular) toric divisor looks like $T^*T^2$, which has a natural Weinstein structure described by a diagram discovered by Gompf \cite{Gompf}.

The toric divisor is the preimage of the facets of the Delzant polytope $\Delta$, and the nodes are the preimages of the vertices of $\Delta$. Since there is a one to one correspondence between the nodes and vertices, we will use the same notation to denote both a node and its moment map image vertex. Each vertex $V\in \Delta$, has a corresponding ray $r$ based at $V$ defined as the sum of the primitive edge vectors of $\Delta$ adjacent to $V$ and pointing outward from $V$.
\begin{definition}\label{def:centered}
A toric manifold with a chosen subset $\{V_1, \ldots, V_n\}$ of the nodes is $\{V_1, \ldots , V_n\}$-centered, if the corresponding rays $r_1, \ldots , r_n$ all intersect at a common single point in the interior of its Delzant polytope.
\end{definition}
We show in \cite[Theorem 4.1]{ACGMMSW} that if a toric $4$-manifold is $\{V_1, \ldots , V_n\}$-centered, then the complement of the toric divisor smoothed at the nodes $\{V_1, \ldots, V_n\}$ has a Weinstein structure which we can explicitly describe.

In this article, we explain an algorithm to produce a Weinstein handlebody diagram for the complement of any divisor obtained by smoothing the $\{V_1, \ldots, V_n\}$-nodes of a toric divisor in a $\{V_1, \ldots, V_n\}$-centered toric $4$-manifold. We prove the Weinstein handlebody produced by this algorithm is Weinstein homotopic to the complement of the smoothed toric divisor in our later paper \cite{ACGMMSW}. That article also explains how toric moment data determines the input to our algorithm, provides a detailed exploration of the centered hypothesis, and includes many more examples.
This article focuses on the most accessible example, as well as showcasing one fun case.

\subsection*{Main Results}
There exists an algorithm to produce a Weinstein handlebody diagram for the complement of a toric divisor smoothed at the subset of nodes $\{V_1, \ldots, V_n\}$ in a $\{V_1, \ldots, V_n\}$-centered toric $4$-manifold.
\begin{enumerate}
\item Applying this algorithm to $\mathbb{C}\P^2$ smoothed in one node yields the self-plumbing of $T^*S^2$ as illustrated in Figure \ref{algorithm2}. Moreover, the same output is obtained for the complement of a toric divisor in \textit{any} toric 4-manifold smoothed in one node.
\begin{figure}[!ht]
\begin{center}
\begin{tikzpicture}
\node[inner sep=0] at (0,0) {\includegraphics[width=8 cm]{figures/algorithm2.pdf}};
\draw[->, line width=0.8mm] (1,0) -- (1.3,0);
\draw[line width=0.8mm] (-1.2,0) -- (-1,0);
\draw (-1,-0.5) rectangle (1,0.5) node[pos=.5] {Algorithm};
\end{tikzpicture}
\caption{}
\label{algorithm2}
\end{center}
\end{figure}

\vspace{-.1in}

\item Applying the algorithm to a monotone $\mathbb{C}\P^2 \# 3\overline{\C\P}\,\!^2$ smoothed at all six nodes yields a Weinstein manifold constructed by attaching $2$-handles to the $4$-ball along a $5$-component Legendrian link as in Figure~\ref{algorithm1}.
\begin{figure}[!ht]
\begin{center}
\begin{tikzpicture}
\node[inner sep=0] at (0,0) {\includegraphics[width=8 cm]{figures/algo.pdf}};
\draw[->, line width=0.8mm] (1,0) -- (1.3,0);
\draw[line width=0.8mm] (-1.2,0) -- (-1,0);
\draw (-1,-0.5) rectangle (1,0.5) node[pos=.5] {Algorithm};
\end{tikzpicture}
\caption{}
\label{algorithm1}
\end{center}
\end{figure}
\end{enumerate}

We would like to note that the handlebody diagram in the first example above has already been observed by Casals-Murphy in \cite{CM}, viewed as a Weinstein handlebody for the complement of an affine smooth conic in $\mathbb{C}^2$ (which is the same as the complement of a smooth conic together with a generic line in $\C\P^2$, which is the once smoothed toric divisor in $\C\P^2$). We obtain this diagram from a completely different method and provide a systematic recipe which applies much more generally. This first example provides an accessible way to explain the steps of our more general algorithm. \smallskip

This paper is organized as follows. In Section 2, we give definitions and discuss the relevant preliminary background on Weinstein Kirby calculus. In Section 3, we provide a picture into how to see the main structure of the handle attachment needed to obtain the complement of the smoothed divisor by describing the core and co-core of the handle in the smoothing local model. The remainder of the paper demonstrates the algorithm for producing the desired handlebody diagrams. In Section~4, we produce a Weinstein handlebody diagram for the complement of a toric divisor smoothed in one node and apply sequences of Kirby calculus moves to simplify the diagrams. In Section~5, we present a more complicated example, coming from $\C\P^2 \# 3\overline{\C\P}\,\!^2$, with the toric divisor smoothed at all six nodes, to showcase the scope of applications and the corresponding Weinstein Kirby diagrams.

\subsection*{Acknowledgments} This project was initiated at the \href{https://icerm.brown.edu/topical_workshops/tw19-4-wiscon/}{2019 Research Collaboration Conference for Women in Symplectic and Contact Geometry and Topology (WiSCon)} that took place on July 22-26, 2019 at ICERM. The authors would like to extend their gratitude to the WiSCon organizers and the hosting institution ICERM and their staff for their hospitality. The authors would like to thank Lenhard Ng for useful conversations. BA and AM would like to thank the Oberwolfach Research Institute for Mathematics for hosting them during the completion of this project. AG was partially supported by Wallenberg grant no. KAW 2016-0440 and the Fondation Math\'ematique Jacques Hadamard. AM is partially supported by Ministry of Education and Science of Republic of Serbia, project ON174034.
LS is supported by NSF grant no. DMS 1904074. AW is supported by the Engineering and Physical Sciences Research Council [EP/L015234/1], the EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London. OC-S is supported by NSF Graduate Research Fellowship under grant no. DGE-1644868.

\section{Weinstein handlebodies and Kirby calculus}

\subsection{Weinstein handle structure}

A \emph{Liouville vector field} $Z$ for a symplectic manifold $(W,\omega)$ is a vector field satisfying $\mathcal{L}_Z \omega = \omega$. By Cartan's formula for the Lie derivative and the fact that the symplectic form is closed, this is equivalent to saying $d(\iota_Z\omega)=\omega$ (here $\iota$ denotes the interior product where $\iota_Z\omega(\cdot) = \omega(V,\cdot)$). In particular, when there exists a Liouville vector field, the symplectic structure is exact. The $1$-form $\lambda = \iota_Z\omega$ which satisfies $d\lambda=\omega$ is called the \emph{Liouville form}. For an introduction to these ideas, see \cite{mcduff2017introduction}. The primary use of Liouville vector fields is to glue symplectic manifolds along contact type boundaries. When the Liouville vector field is transverse to the boundary, it defines a contact structure on the boundary and can be used to identify a collared neighborhood of the boundary with a piece of the symplectization of that contact boundary.\smallskip

When Weinstein defined a model of a handle decomposition for symplectic manifolds \cite{Weinstein}, he equipped the handle with a Liouville vector field so that the gluing of the handle attachment could be performed using only contact information on the boundary. More specifically, the handle attachment is completely specified by a Legendrian attaching sphere, or an isotropic attaching sphere together with data on its normal bundle. The limitation is that the index of the handle is required to be less than or equal to $n$ in a $2n$ dimensional manifold. In particular, Weinstein $4$-manifolds must be built entirely from handles of index $0$, $1$, and $2$. \smallskip

The model Weinstein handle of index $k$ in dimension $2n$ for $k\leq n$ is a subset of $\mathbb{R}^{2n}$ with coordinates $(x_1,y_1,\cdots, x_n,y_n)$, with the standard symplectic structure $\omega = \sum_j dx_j\wedge dy_j$ and Liouville vector field
$$Z_k = \sum_{j=1}^k  \left( -x_j\partial_{x_j}+2y_j\partial_{y_j} \right) + \sum_{j={k+1}}^n  \left(\frac{1}{2}x_j\partial_{x_j}+\frac{1}{2} y_j\partial_{y_j}\right).$$

\begin{figure}
	\centering
	\includegraphics[scale=.5]{figures/handle}
	\caption{A sketch of the model for the Weinstein handle of index $k$, with the associated gradient-like Liouville vector field $Z_k$.}
	\label{fig:model}
\end{figure}

As with smooth handle theory, the handles are in one to one correspondence with the critical points of a Morse function. The Liouville vector field agrees with the gradient of such a Morse function (for some choice of metric), in other words, the Liouville vector field is \emph{gradient-like}. In the model index $k$ handle, the Liouville vector field is the gradient (with the standard Euclidean metric) of the function
$$\phi_k = \sum_{j=1}^k \left(-\frac{1}{2} x_j^2 + y_j^2\right) + \sum_{j={k+1}}^n \left(  \frac{1}{4}x_j^2+\frac{1}{4}y_j^2 \right).$$

The handle can be considered to be the subset of $\mathbb{R}^{2n}$ given by $D^k\times D^{2n-k}$ where the first factor corresponds to the coordinates $(x_1,\cdots, x_k)$ and the second corresponds to the remaining coordinates $(x_{k+1},\cdots, x_n,y_1,\cdots, y_n)$. The key terminology for important parts of the handle is as follows.

\begin{itemize}
	\item The \emph{core} of the handle is $D^k\times \{0\}$ where $x_{k+1}=\cdots=x_n=y_1=\cdots=y_n=0$. This is the \emph{stable manifold} of flow-lines of $Z_k$ which limit positively towards the zero at the origin.
	\item The \emph{co-core} of the handle is $\{0\}\times D^{2n-k}$ where $x_1=\cdots=x_k=0$. This is the \emph{unstable manifold} of flow-lines of $Z_k$ which limit negatively towards the zero at the origin.
	\item The \emph{attaching sphere} is the boundary of the core, $S^{k-1}\times\{0\}$. This will be identified with an isotropic sphere in the boundary of the existing manifold to which the handle is attached.
	\item The \emph{attaching region} is a neighborhood of the attaching sphere $S^{k-1}\times D^{2n-k}$. This is the entire part of the handle which will be glued on to a piece of the boundary of the existing manifold when the handle is attached. Therefore the Liouville vector field $Z_k$ points inward into the handle along this part of the boundary (it is \emph{concave}).
	\item The \emph{belt sphere} is the boundary of the co-core $\{0\}\times S^{2n-k-1}$. It is a co-isotropic sphere in the boundary of the manifold obtained after attaching the handle.
\end{itemize}

In general we can piece together the Liouville vector fields on the handles, and put together adjusted versions of the locally defined Morse functions to get a global Morse function on the manifold. A Weinstein structure is often encoded analytically as a quadruple $(W,\omega, Z, \phi)$ where $W$ is a smooth manifold, $\omega$ is a symplectic structure on $W$, $Z$ is a Liouville vector field for $\omega$ on $W$, and $\phi$ is a Morse function such that $Z$ is the gradient-like for $\phi$.

\begin{remark}
	When a manifold with a Weinstein structure has contact type boundary, it is called a \emph{Weinstein domain}. Such a domain can be extended by a cylindrical end to make the Liouville vector field complete to give a non-compact infinite volume \emph{Weinstein manifold}.
\end{remark}

\subsection{Weinstein Kirby calculus} \label{s:kirby}

The data needed to encode the Weinstein domain are the attaching maps. In dimension $4$, the attaching map of a handle is completely determined by the Legendrian or isotropic attaching sphere. The attaching sphere of a $1$-handle is a pair of points. Diagramatically we draw a pair of $3$-balls implicitly identified by a reflection, representing the attaching region $S^0\times D^3$. The attaching sphere of a $2$-handle is a Legendrian embedded circle (a Legendrian knot). The $4$-dimensional $2$-handle attachment is determined by the knot together with a framing, but in the Weinstein case, the framing is determined by the contact structure. More specifically, the contact planes along a Legendrian knot determine a framing by taking a vector field transverse to the contact planes. The contactomorphism gluing the attaching region of the $2$-handle to the neighborhood of the Legendrian identifies the product framing in the $2$-handle with the $tb-1$ framing. Here $tb$ denotes the contact framing (Thurston-Bennequin number), which is identified with an integer by looking at the difference between the contact framing and the Seifert framing (this must be appropriately interpreted when the diagram contains $1$-handles--see \cite{Gompf}). \smallskip

The diagram we draw should specify the Legendrian attaching knots in $S^3$ along with the pairs of $3$-balls indicating the attachments of the $1$-handles. By removing a point away from these attachments, we reduce the picture in $S^3$ to a picture in $\mathbb{R}^3$. After a contactomorphism, the contact structure on $\mathbb{R}^3$ is $\ker(dz-ydx)$ in coordinates $(x,y,z)$. The \emph{front projection} is the map $\pi: \mathbb{R}^3\to \mathbb{R}^2$ with $\pi(x,y,z)=(x,z)$. A Legendrian curve in this contact structure is tangent to the contact planes, which happens precisely when the $y$-coordinate is equal to the slope $\frac{dz}{dx}$ of the front projection. Therefore, Legendrian knots can be recovered from their front projections with the requirement that the diagram has no vertical tangencies (instead it will have cusp singularities where the knot is tangent to the fibers of the projection) and the crossings are always resolved so that the over-strand is the strand with the more negative slope (we orient the $y$-axis into the page to maintain the standard orientation convention for $\mathbb{R}^3$ so the over-strand is the strand with a more negative $y$-coordinate). In these front projections, the contact framing $tb$ can be computed combinatorially in terms of the oriented crossings and cusps of the diagram when the diagram is placed in a standard form where the pairs of $3$-balls giving the attaching regions of $1$-handles are related by a reflection across a vertical axis. Namely, $tb$ of a Legendrian knot is the difference of the writhe of the knot and half the number of cusps in the front projection. For a thorough introduction to Legendrian knots, see \cite{ETNYRE2005105}.\smallskip

The set of moves that relate Weinstein handlebody diagrams in Gompf standard form for equivalent Weinstein domains includes Legendrian Reidemeister moves (including how they interact with the $1$-handles) listed in \cite{Gompf}, see Figures \ref{Reid} and \ref{Gompf456}, as well as \textit{handle slides}, and \textit{handle pair cancellations and additions}, shown in \cite{DingGeiges}. Given two $k$-handles, $h_1$ and $h_2$, a \textit{handle slide} of $h_1$ over $h_2$ is given by isotoping the attaching sphere of $h_1$, and pushing it through the belt sphere of $h_2$. We depict a 1-handle slide (along with intermediate Reidemeister and Gompf moves) in Figure \ref{1hslide} and a 2-handle slide in Figure \ref{2hslide}. A 1-handle $h_1$ and a 2-handle $h_2$ can be cancelled, provided that the attaching sphere of $h_2$ intersects the belt sphere of $h_1$ transversely in a single point. We call this a \textit{handle cancellation} and the pair of handles a \textit{cancelling pair}. Likewise a cancelling pair can be added to a Weinstein handlebody diagram, as depicted in Figure \ref{2hcancel}. When multiple 2-handles intersect a single 1-handle, the simplification in Figure \ref{2handles} can be performed to reduce the overall complexity of a Weinstein diagram.\smallskip

\begin{figure}[h!]
	\begin{center}
		\includegraphics[width=8cm]{figures/Reid.pdf}{}
		\caption{The Legendrian Reidemeister moves, up to 180 degree rotation about each axis, where the top, middle, and bottom moves are called Reidemeister I, Reidemeister II, and Reidemeister III, respectively.}
		\label{Reid}
	\end{center}
\end{figure}

\begin{figure}[h!]
	\begin{center}
		\includegraphics[width=12cm]{figures/Gompf456.pdf}
		\caption{Gompf's three additional isotopic moves, up to 180 degree rotation about each axis. The top, middle, and bottom moves are called Gompf move 4, Gompf move 5, and Gompf move 6, respectively.}
		\label{Gompf456}
	\end{center}
\end{figure}

\begin{figure}[h!]
	\begin{center}{}
	\begin{tikzpicture}
			\node[inner sep=0] at (0,0) {\includegraphics[width=12cm]{figures/TstarT21hslide.pdf}};
			\node at (-2.05,1.3){\footnotesize (1)};
			\node at (2.05,1.28){\footnotesize (2)};
			\node at (0.5,0.5){\footnotesize (3)};
			\node at (-2.05,-1.7){\footnotesize (4)};
			\node at (2.05,-1.65){\footnotesize (5)};
		\end{tikzpicture}
		\caption{An example of a 1-handle slide on $T^*T^2$ consisting of the following sequence of isotopies: (1) Reidemeister 2, (2) Gompf move 4, (3) Gompf move 5, (4) Gompf move 4, (5) Reidemeister 2.}
		\label{1hslide}
	\end{center}
\end{figure}

\begin{figure}[h!]
	\begin{center}
		\includegraphics[width=7cm]{figures/2hslide.pdf}
		\caption{An example of a 2-handle slide of the black unknot over the red unknot.}
		\label{2hslide}
	\end{center}
\end{figure}{}

\begin{figure}[h!]
	\begin{center}
		\includegraphics[width=11cm]{figures/2hcancel.pdf}
		\caption{An example of a 1-handle cancelling with a 2-handle.}
		\label{2hcancel}
	\end{center}
\end{figure}{}

\begin{figure}[h!]
	\begin{center}
		\includegraphics[width=8cm]{figures/2handleslides.pdf}
		\caption{An example of handle slides and cancellations when multiple 2-handles pass through a 1-handle. Red and blue 2-handles are slid over the central green 2-handle. The green 2-handle is then cancelled with the 1-handle.}
		\label{2handles}
	\end{center}
\end{figure}{}

Before approaching our goal of presenting an algorithm to construct Weinstein--Kirby diagrams for complements of smoothed toric divisors, we will start with the unsmoothed case, where the complement has a Liouville completion, $T^*T^2$. The Legendrian handlebody we present, was originally found by Gompf \cite{Gompf}. It follows from that article that this handlebody gives a Stein/Weinstein structure on the smooth manifold $D^*T^2$ (which is the trivial bundle $D^2\times T^2$). More generally, Stein handlebody diagrams are given on the smooth manifolds $D^*\Sigma$ for any surface in \cite{Gompf}. In \cite[Theorem 7.1]{ACGMMSW}, we show that the Weinstein structures induced on these diagrams are Weinstein homotopic to the canonical co-tangent Weinstein structure on $D^*\Sigma$.

For $T^*T^2$ specifically, it is known that there is a unique Weinstein fillable contact structure on the boundary $T^3$ and one can then deduce that the Gompf handlebody agrees with the canonical symplectic structure by Wendl's result that $S^*T^2$ has a unique Stein/Weinstein filling up to deformation \cite{Wendl}. To see that the diagram in Figure \ref{TstarT2} represents $D^*T^2\cong D^2\times T^2$ smoothly, we can start with a handle decomposition for $T^2$ with one 0-handle, two 1-handles and a single 2-handle. Thickening this diagram to a $4$-dimensional handlebody yields a disk bundle over $T^2$ with Euler number $e$, agreeing with the framing coefficient of the $2$-handle attachment. One then needs to put the diagram into Gompf standard form as seen in Figure \ref{stdTstarT2} by sliding the uppermost attaching ball below the attaching ball on the right so that both $1$-handles are related by a reflection across the same vertical axis. Then we must realize the knot as a Legendrian knot by replacing vertical tangencies by cusps and making sure the crossings always have the over-strand corresponding to the more negative slope. The most obvious way to do this yields a Legendrian knot whose Thurston-Bennequin framing is $0$ (see the diagram on the right of Figure \ref{stdTstarT2}), so this would correspond to a $D^2$ bundle over $T^2$ with Euler number $-1$. By wrapping one strand around the lower left attaching ball as in Figure \ref{TstarT2}, we obtain a smoothly isotopic picture where the new Legendrian has $tb=1$, so the Euler number is $1-1=0$ as needed for $D^*T^2$. Since this is one Weinstein filling of $S^*T^2$, and we know that such fillings are unique up to deformation, it must agree with the canonical co-tangent symplectic structure on $D^*T^2$.

\begin{figure}[h!]
    \centering
    \subfigure{\includegraphics[width=0.25\textwidth]{figures/basicslide.pdf}} \hspace{2.5cm}
    \subfigure{\includegraphics[width=0.30\textwidth]{figures/stdform.pdf}}

    \caption{Diagram depicting how to move the usual picture of $T^*T^2$ into standard form after inserting the necessary Legendrian data, i.e. replacing vertical tangencies by cusps.}
    \label{stdTstarT2}
\end{figure}

\begin{figure}[h!]
	\begin{center}
		\includegraphics[width=6cm]{figures/TstarT2.pdf}
		\caption{Stein structure on $D^2$ bundle over $T^2$ with framing coefficient $e(T^*T^2) \leq 0$.}
		\label{TstarT2}
	\end{center}
\end{figure}

\section{The local model for our handle attachment} \label{s:handle}

In this section we study the local model for the smoothing of a normal crossing singularity of a symplectic
divisor in dimension~$4$ and describe the local handle attachment information (core and co-core) for the $2$-handle attachment needed to describe the complement of the smoothing of a node. Here we give the intuitive picture behind the following theorem.

\begin{theorem}[Theorem 4.1~\cite{ACGMMSW}]\label{thm:toricWein}
	Let $(M,\omega)$ be a toric $4$-manifold corresponding to Delzant polytope $\Delta$ which is $\{V_1,\ldots, V_n \}$-centered. Let $D$ denote the divisor obtained by smoothing the toric divisor at the nodes $V_1,\ldots, V_n$. Then there exist arbitrarily small neighborhoods $N$ of $D$ such that $M\setminus N$ admits the structure of a Weinstein domain.

	Furthermore, $M\setminus N$ is Weinstein homotopic to the Weinstein domain obtained by attaching Weinstein $2$-handles to the unit disk cotangent bundle of the torus $D^*T^2$, along the Legendrian co-normal lifts of co-oriented curves of slope $s(V_1),\ldots, s(V_n)$. Here $s(V_i)$ is equal to the difference of the inward normal vectors of the edges adjacent to $V_i$ in $\Delta$.
\end{theorem}

A local model in a $4$-dimensional manifold~$M$ for the normal crossing intersection of two
symplectic divisors can be given by a Darboux chart~$(\mathbb{C}^2, \omega_{std})$ at the
intersection point, where the two divisors are mapped to the two axes
$\Sigma_1 = \left \{{(z_1,z_2) \in \mathbb{C}^2 \, | \, z_2 = 0} \right\}$ and
$\Sigma_2 = \left \{{(z_1,z_2) \in \mathbb{C}^2 \, | \, z_1 = 0} \right\}$.
Smoothing this normal crossing means that locally one substitutes the union of these divisors
by the smooth surface
$$\Sigma = \left \{{(z_1,z_2) \in \mathbb{C}^2  \, | \,  z_1 \cdot z_2 = \varepsilon^2} \right\}$$
for some $\varepsilon > 0$. Topologically, the complement of the smoothed surface differs from the complement of the normal crossing divisors by one 2-handle attachment for each smoothed normal crossing intersection. Under the centered hypothesis (Definition~\ref{def:centered}), Theorem~\ref{thm:toricWein} says that when considering the symplectic and Liouville structures, the difference between the complements also corresponds to a collection of \emph{Weinstein} 2-handle attachments.

In order to encode this handle attachment in the Weinstein Kirby diagram, we need to
identify the corresponding Legendrian attaching sphere. We find natural Lagrangian representatives of the co-core and core of the handle in the complement of the smoothed divisor. From there, the attaching sphere is the boundary of the core.

A co-core of a Weinstein $2$-handle is characterized (uniquely up to Lagrangian isotopy \cite{EliPol}) as a Lagrangian disk contained in the smooth $2$-handle with unknotted Legendrian boundary lying in the boundary of the handle (not the attaching region). In our model, the boundary of the handle is in the boundary of the small neighborhood of the smoothed hypersurface $\Sigma$.
One can consider, for $0 < \varepsilon' < \varepsilon$, the disk $D_1$ defined as the image of the map
$$
\begin{array}{ccccc}
\phi & : & [0,1] \times [0,2\pi] & \rightarrow & \mathbb{C}^2\\
&   & (r,\theta)                   & \mapsto    & \left({ \begin{array}{c}
	r \varepsilon' e^{i \theta}\\
	r \varepsilon' e^{-i \theta}
	\end{array}}\right)
\end{array}
$$
The size of $\varepsilon'$ is determined by the size of the neighborhood of the smoothed divisor which is deleted. As $\varepsilon'\to \varepsilon$, the open disk of radius $\varepsilon$ provides the analog of the co-core in the non-compact complement of the divisor itself.

This disk is Lagrangian as for $r \neq 0$, the derivative of $\phi$
$$d_{(r,\theta)} \phi = \left({
	\begin{array}{cc}
	\varepsilon' e^{i \theta} & i r \varepsilon' e^{i \theta}\\
	\varepsilon' e^{-i \theta} & -i r \varepsilon' e^{-i \theta}
	\end{array}
}\right)
$$
is an isomorphism so that the tangent space to $D_1$  at the point $\phi(r,\theta)$ is spanned
by the vectors
$$u_1 = \left({ \begin{array}{r}
	\varepsilon' e^{i \theta}\\
	\varepsilon' e^{-i \theta}
	\end{array}}\right)
\textrm{ and }
u_2 = \left({ \begin{array}{r}
	i r \varepsilon' e^{i \theta}\\
	-i r \varepsilon' e^{-i \theta}
	\end{array}}\right)$$
and one can check that
$$\omega_{std}(u_1, u_2) = \Im m ( \langle u_1 ,  u_2 \rangle )  = 0$$
where $\langle \cdot ,  \cdot \rangle$ is the standard Hermitian product in $\mathbb{C}^2$
(equivalently one can check that $\phi^*(\frac{i}{2}(dz_1\wedge d\bar{z}_1+dz_2\wedge d\bar{z}_2))=0$). The point for $r=0$ corresponds to the origin of $\mathbb{C}^2$. At this point, the two following curves $c_1$
and $c_2$ in $D_1$ parametrized for $t \in (-1,1)$ by:
$$c_1(t) = \left({ \begin{array}{c}
	t \varepsilon'\\
	t \varepsilon'
	\end{array}}\right)
\textrm{ and }
c_2(t) = \left({ \begin{array}{c}
	i t \varepsilon'\\
	-i t \varepsilon'
	\end{array}}\right)$$
give the two independent vectors in the tangent space at the origin of the disk~$D_1$:
$$u_1 = \left({ \begin{array}{c}
	\varepsilon' \\
	\varepsilon'
	\end{array}}\right)
\textrm{ and }
u_2 = \left({ \begin{array}{c}
	i  \varepsilon' \\
	-i  \varepsilon'
	\end{array}}\right)$$
One can note that we have again $\omega_{std}(u_1, u_2)  = 0$.

The boundary of this disk is the image by $\phi$ of $\{1\} \times [0,2\pi],$ that is, the circle
$$B = \left\{ { \left. {\left({ \begin{array}{c}
			\varepsilon' e^{i \theta}\\
			\varepsilon' e^{-i \theta}
			\end{array} }\right) } \right| \theta \in [0, 2\pi] }\right\}.$$

One can choose $\varepsilon'<\varepsilon$ such that the circle $B$ lies on the boundary of the small neighborhood of the smoothed $\Sigma$ (since $z_1 z_2 = \varepsilon' e^{i \theta} \varepsilon' e^{-i \theta} = \varepsilon'^2$ goes to $\varepsilon^2$ if $\varepsilon'$ approaches $\varepsilon$). It limits to the origin when $\varepsilon'$ goes to $0$, so that $B$ is the belt sphere of the handle and $D_1$ is indeed its co-core.\smallskip

The core is characterized (uniquely up to Lagrangian isotopy) as a Lagrangian disk with unknotted boundary in the smooth $2$-handle which intersects the co-core transversally at one point and which avoids the boundary of the Weinstein manifold, so in our model, it should avoid the smoothed~$\Sigma$.

Let $D_2$ be the disk defined as the image of the map
$$
\begin{array}{ccccc}
\psi & : & [0,1] \times [0,2\pi] & \rightarrow & \mathbb{C}^2\\
&   & (r,\theta)                   & \mapsto    & \left({ \begin{array}{c}
	r \varepsilon e^{i \theta}\\
	r \varepsilon e^{-i (\theta+\pi)}
	\end{array}}\right)
=  \left({ \begin{array}{c}
	r \varepsilon e^{i \theta}\\
	-r \varepsilon e^{-i \theta}
	\end{array}}\right)
\end{array}
$$

This disk is also Lagrangian as,
similarly as before,
the tangent space to $D_2$  at the point $\psi(r,\theta)$ for $r \neq 0$ is spanned
by the vectors
$$u_3 = \left({ \begin{array}{r}
	\varepsilon e^{i \theta}\\
	-\varepsilon e^{-i \theta}
	\end{array}}\right)
\textrm{ and }
u_4 = \left({ \begin{array}{r}
	i r \varepsilon e^{i \theta}\\
	i r \varepsilon e^{-i \theta}
	\end{array}}\right)$$
and one can check that $$\omega_{std}(u_3, u_4) = \Im m ( \langle u_3 ,  u_4 \rangle )  = 0.$$

Similarly, at the origin, the two following curves $c_3$ and $c_4$ in $D_2$ parametrized for $t \in (-1,1)$ by
$$c_3(t) = \left({ \begin{array}{c}
	t \varepsilon\\
	-t \varepsilon
	\end{array}}\right)
\textrm{ and }
c_4(t) = \left({ \begin{array}{c}
	i t \varepsilon\\
	i t \varepsilon
	\end{array}}\right)$$
give the two independent vectors in the tangent space at the origin of the disk~$D_2$:
$$u_3 = \left({ \begin{array}{c}
	\varepsilon \\
	-\varepsilon
	\end{array}}\right)
\textrm{ and }
u_4 = \left({ \begin{array}{c}
	i  \varepsilon \\
	i  \varepsilon
	\end{array}}\right)$$
Note again that $\omega_{std}(u_3, u_4)  = 0$. This disk does not intersect the smoothed  $\Sigma$ as
$$z_1 z_2 = r \varepsilon e^{i \theta} (- r \varepsilon e^{-i \theta}) = - r^2 \varepsilon^2 \neq \varepsilon^2.$$ Moreover, $D_1$ and $D_2$ intersect at the origin and this intersection is transverse as one can check that the family $(u_1, u_2, u_3, u_4)$ spans $\mathbb{C}^2$ as a real vector space. This shows that $D_2$ is the core of the handle and the attaching sphere is  the image by $\psi$ of $\{1\} \times [0,2\pi]$, that is, $$A = \left\{ { \left. {\left({ \begin{array}{c}
			\varepsilon e^{i \theta}\\
			\varepsilon e^{-i (\theta+\pi)}
			\end{array} }\right) } \right| \theta \in [0, 2\pi] }\right\}.$$

Identifying these transverse Lagrangian disks with the core and co-core in Weinstein's model for a $4$-dimensional $2$-handle, allows us to place a Weinstein structure (a Liouville vector field and corresponding Morse function) on this new piece which lies in the complement of the regular neighborhood of the smoothing. If the polytope is centered with respect to all the nodes we are smoothing, the Weinstein structures on these pieces glue together consistently with a Weinstein structure on the complement of the unsmoothed toric divisor (a Weinstein domain which completes to $T^*T^2$). It is shown in~\cite[Section 4.5]{ACGMMSW} how these fit together to give an explicit global Weinstein structure.

In the toric description of the toric manifolds and divisors we consider, the Hamiltonian torus action in the local Darboux model corresponds to the torus action on $\mathbb{C}^2$ given in coordinates: $(e^{i\theta_1}, e^{i\theta_2}) \ast (z_1,z_2)=(e^{i\theta_1}z_1,e^{i\theta_2}z_2)$. In particular, through the symplectomorphism between the complement of the normal crossing divisor we consider and~$T^*T^2$, the orbit of a point corresponds to the torus
$T^2$ and the quotient space under the Hamiltonian action corresponds to a cotangent
fiber. The attaching sphere in the model corresponds in this symplectic identification
to a lift of a circle of slope~$(1, -1)$ in the base~$T^2$ to the
cotangent bundle~$T^*T^2$ (lift corresponding to the
point~$(\varepsilon^2, \varepsilon^2)$ in the quotient space $(\mathbb{R}_{>0})^2$).
The standard model neighborhood corresponds to the standard cone $\mathbb{R}_{\geq 0}\times \mathbb{R}_{\geq 0}$, and the corresponding attaching slope is $(1,-1)=(1,0)-(0,1)$, the difference of the inward normal vectors.
In general, relating any node to the standard model via an $SL(2,\mathbb{Z})$ transformation, the attaching sphere for the $2$-handle corresponding to the smoothing of a normal crossing singularity at a chosen vertex $V$ will be the co-normal lift of a circle of slope $s(V)$, where $s(V)$ is the difference of the inward normal vectors of the edges adjacent to $V$,
(see~\cite[Lemma 4.2]{ACGMMSW}).

\section{The algorithm through an example}

Here we will show how to obtain a Weinstein handle diagram for the complement of a toric divisor with exactly one node smoothed. Note that for any Delzant polytope, and any single vertex $V$, the polytope is vacuously $\{V\}$-centered. To fix a specific example, we could start with the toric $4$-manifold $\C\P^2$ whose toric divisor is a collection of three $\C\P^1$'s intersecting at three points, and smooth this divisor at one of the intersection points. More explicitly in homogeneous coordinates $[z_0:z_1:z_2]$ on $\C\P^2$, the toric divisor is given by the union of the three lines $L_0=\{z_0=0\}$, $L_1=\{z_1=0\}$ and $L_2=\{z_2=0\}$. Let us smooth the intersection of $L_1$ with $L_2$ at $[1:0:0]$. In the affine coordinate chart where $z_0=1$, with coordinates $(z_1,z_2)$, this aligns exactly with our local model in Section \ref{s:handle}. After smoothing, the lines $L_1$ and $L_2$ are joined to form a conic $Q$, which intersects the remaining line $L_0$ at two points. By Theorem~\ref{thm:toricWein}, the complement of this smoothed divisor is obtained by attaching a single $2$-handle to $D^*T^2$ along the Legendrian lift of a curve in $T^2$ of slope $(1,-1)$.

	\begin{figure}[!ht]
		\centering
		\includegraphics[scale=.3]{figures/torushandlebody1}
		\caption{Left: The Legendrian unknot $\mathcal{K}$ in the boundary of the $4$-dimensional $0$-handle of the Gompf diagram, indicating the intersection of this boundary with the Lagrangian torus ($0$-section of $D^*T^2$). The black portions coincide with segments of the attaching circle of the $2$-handle, and the blue portions give the attaching arcs of the $2$-dimensional $1$-handles of the torus. Right: The corresponding decomposition of the Lagrangian torus.}
		\label{fig:torusdecomp}
	\end{figure}

	In order to translate Legendrian attaching circles in $S^*T^2$ described as the co-normal lift of a curve in $T^2$ into Legendrian curves drawn in the Gompf diagram (Figure \ref{TstarT2}), we need to understand how these two pictures get identified. As mentioned in Section \ref{s:kirby}, the Gompf handle diagram is obtained by starting with a smooth handle decomposition of $T^2$ with a single $0$-handle, two $1$-handles, and one $2$-handle. This diagram is thickened by two dimensions to obtain $T^2\times D^2$, and then the attaching curve of the $2$-handle is isotoped around until it agrees with a Legendrian front diagram with induced framing $tb-1 = 0$. On the other hand, the co-normal lift construction is more compatible with the canonical (Morse-Bott) Weinstein structure on $D^*T^2$ which has critical locus along the $0$-section. In \cite[Theorem 7.1]{ACGMMSW}, we prove that these two structures are Weinstein homotopic and identify the image of the Lagrangian torus giving the zero-section of $D^*T^2$ in the Gompf diagram. The handle decomposition on the $4$-manifold induces the corresponding handle decomposition on the Lagrangian torus by intersection. In particular, we see a Legendrian (un)knot $\mathcal{K}$ in the boundary of the $4$-dimensional $0$-handle, which partially coincides with the attaching sphere of the $2$-handle, and partially corresponds with attaching arcs for the $1$-handles of the Legendrian torus. See Figure \ref{fig:torusdecomp}.

\begin{figure}[!ht]
	\begin{center}
		\includegraphics[width=8cm]{figures/onenodetorus.pdf}
		\caption{The circle with $(1,0)$-slope on $T^2$ with the dotted arrow indicating the co-orientation of the curve, and an isotoped version which runs parallel to the boundary of the $0$-handle of $T^2$ except where it passes through a $1$-handle.}
		\label{onenodetorus}
	\end{center}
\end{figure}

\begin{figure}[!ht]
	\begin{center}
		\includegraphics[width=10 cm]{figures/onenodejet.pdf}
		\caption{The curve in $J^1(S^1)$ which is identified with the curve $(1,0)$ on $T^2$ as in Figure \ref{onenodetorus}.}
		\label{onenodejet}
	\end{center}
\end{figure}

\begin{figure}[!ht]
    \centering
    \subfigure[The neighborhood $U$ of the co-normal lift of a curve $\gamma \subset T^2$]{\includegraphics[width=0.17\textwidth]{figures/square_gradient_corrected}}  \hspace{2cm}
    \subfigure[Identification of $U$ with neighborhood of the zero section of $J^1(S^1)$.]{\includegraphics[width=0.32\textwidth]{figures/jet_space_gradient_corrected}}
\vspace{1cm}
    \subfigure[The front projection of $U$ in the Gompf diagram for $D^*T^2$]{\includegraphics[width=0.43\textwidth]{figures/gradient_corrected}}
\caption{Transferring curves from $T^2$ to $J^1(S^1)$ and then to the Gompf diagram.}
\label{fig:gradient}
\end{figure}

	Now consider the Legendrian co-normal lift to $S^*T^2$ of the circle in $T^2$ which is the boundary of the $2$-dimensional $0$-handle, with the inward co-orientation. This is a Legendrian push-off of $\mathcal{K}$ in $\partial B^4 = \partial D^*D^2$, because a small positive Reeb flow applied to $\mathcal{K}=\partial D^2$ yields the co-normal lift of a concentric circle close to $\partial D^2$. We will perform an isotopy to the curves in our torus corresponding to attaching spheres of the additional $2$-handles so that these curves agree with parallel copies of such circles except where they enter the $1$-handles. Circles which are further inward will be pushed off more in the \emph{positive} Reeb direction. Note that every Legendrian circle has a standard neighborhood which is contactomorphic to a neighborhood of the zero section in $J^1(S^1)$ with the contact form $dz-ydx$ where $x$ is the coordinate on $S^1$. We will translate the diagram on the torus to a front projection diagram of $J^1(S^1)$, where we think of the $S^1$ as the Legendrian boundary of the $0$-handle of the $T^2$ $0$-section. Since the Reeb direction is the positive $\partial_z$ direction in $J^1(S^1)$, circles which are further inwards in the torus (pushed further by the Reeb flow) will correspond to curves which are pushed upwards more in the $J^1(S^1)$ diagram. See an example of this procedure in Figures \ref{onenodetorus} and \ref{onenodejet}. Finally, once we have our diagram in the 1-jet space, we can satellite the diagram onto the image of $S^1$ in the Gompf diagram. The images of the co-normal lifts of curves contained in a neighborhood $U$ of the curve $\gamma$ that coincides with the attaching circle of the $2$-handle and the attaching arcs of the $2$ dimensional $1$ handles of the torus are illustrated in Figure \ref{fig:gradient}. As indicated by the shading gradient, curves which lie further inward towards the center of the square correspond to curves at greater $z$-height values in the jet-space. In turn, curves at a higher $z$-height in the jet space will correspond to higher Reeb pushoffs of the complicated looking Legendrian unknot of Figure~\ref{fig:gradient}(c).

	Initially, let us apply this procedure in the simple example where we are attaching a $2$-handle along the co-normal lift of a circle in the torus with slope $(1,0)$. The cotangent projection of this model is presented on the left in Figure \ref{onenodetorus}. Push the curve $(1,0)$ to the upper side of the square, so it lies close to the boundary, and then cut the rectangle at the bottom left vertex to map to $J^1(S^1)$, obtaining Figure \ref{onenodejet}. Satelliting this onto the Legendrian unknot in Figure \ref{fig:torusdecomp}, we obtain Figure \ref{onenode}.\smallskip

\begin{figure}[!ht]
	\begin{center}
		\includegraphics[width=6cm]{figures/onenode.pdf}
		\caption{The Legendrian handle diagram of the complement of the toric divisor smoothed in one node. That is, $T^*T^2 \cup \Lambda_{(1,0)}$.}
		\label{onenode}
	\end{center}
\end{figure}

	\begin{figure}[!ht]
		\begin{center}
			\includegraphics[width=8cm]{figures/curvecorner.pdf}
			\caption{Mapping the red, blue and purple curves from $J^1(S^1)$ to $T^*T^2$. The relative heights of these curves with respect to the black curve are preserved. The purple curve follows a Reeb push off of the cusping curve in the 1-handle's attaching region (see Figure \ref{fig:torusdecomp}) while the red and blue curves pass through the 1-handle.}
			\label{curvecorner}
		\end{center}
	\end{figure}

	In general, when satelliting, we need to be somewhat careful with the behavior of the curves near the $1$-handle. If the curves pass above the attaching region of a $1$-handle without entering the $1$-handle, they will follow an upward Reeb push-off of the cusps that appear inside the $1$-handle attaching balls in Figure \ref{fig:torusdecomp}. Note that we will typically push these cusps out of the attaching regions of the $1$-handles by a Legendrian isotopy. If the curves pass through the $1$-handle in the torus, they will pass through the corresponding $1$-handle in the $4$-dimensional handlebody. See Figure \ref{curvecorner} for the conventions in a more complicated example.\smallskip

\begin{figure}[!ht]
	\begin{center}
		\includegraphics[width=14cm]{figures/onenodeslide.pdf}
		\caption{In columns from left to right, a series of Reidemeister and Gompf moves and $1$-handle slides that take the Legendrian lift of a $(1,-1)$ curve to the Legendrian lift of a $(1,0)$ curve.  $(1)$ Reidemeister III, $(2)$ Reidemeister II and I, $(3)$ Reidemeister II, $(4)$ Reidemeister II, $(5)$ slide green $1$-handle over orange $1$-handle, $(6)$ Gompf move $5$, $(7)$ Gompf move $4$, and $(8)$ Reidemeister II.}
		\label{onenodeslide}
	\end{center}
\end{figure}

\begin{figure}
\begin{overpic}[scale=.25]{figures/onenodewithmoves.pdf}
\put(35, 93){$(1)$}
\put(35, 83){$(2)$}
\put(35, 70){$(3)$}
\put(35, 57){$(4)$}
\put(35, 47){$(5)$}
\put(35, 35){$(6)$}
\put(35, 26){$(7)$}
\put(35, 15){$(8)$}
\put(35, 6){$(9)$}
\end{overpic}
\caption{A series of Reidemeister moves and handle slides simplifying the Weinstein handle diagram of the complement of the toric divisor smoothed in one node. $(1)$ Reidemeister III, $(2)$ Reidemeister II and I, $(3)$ $2$-handle slide, $(4)$ Handle cancellation, $(5)$ Reidemeister III, $(6)$ Reidemeister I and II, $(7)$ Reidemeister III, I and II, $(8)$ Gompf move 6, $(9)$ Reidemeister II.}
\label{onenodeseries}
\end{figure}

	Although our initial explicit example asked us to attach a handle along a curve of slope $(1,-1)$, we can see that in fact the resulting Weinstein manifold is equivalent to using the $(1,0)$ slope. Figure~\ref{onenodeslide} shows the series of $1$ handle slides, Reidemeister and Gompf moves that take the Legendrian lift of the resulting diagram obtained by attaching along a $(1,-1)$ curve to the diagram corresponding to attaching along a $(1,0)$ curve. \smallskip

\begin{figure}[!ht]
	\begin{center}
		\includegraphics[width=8 cm]{figures/onenodefinal.pdf}
		\caption{The diagram of the complement of the toric divisor smoothed in one node after simplifications, in particular after applying a single Reidemeister I to the last diagram in Figure \ref{onenodeseries}.}
		\label{onenodefinal}
	\end{center}
\end{figure}

\begin{prop}
	For any toric $4$-manifold, let $\Sigma$ be the result of smoothing the toric divisor at a single node. Then the complement of a small regular neighborhood of $\Sigma$ is a Weinstein domain which is Weinstein homotopic to the domain represented by the handle diagram of Figure~\ref{onenode}.
\end{prop}
\begin{proof}
First note that for a single node, the centered hypothesis is automatically satisfied. By Theorem~\ref{thm:toricWein} such a complement is a Weinstein domain obtained by attaching a single 2-handle to $D^*T^2$ along the Legendrian curve given by a co-oriented co-normal lift of a curve of slope $(a,b)$ in the torus. We verify that the resulting Weinstein domain does not depend on the choice of the slope up to Weinstein homotopy. We can see that $T^*T^2 \cup \Lambda_{(a,b)}$ is Weinstein homotopic to $T^*T^2 \cup \Lambda_{(1,0)}$, by performing 1-handle slides on $T^*T^2 \cup \Lambda_{(a,b)}$ similar to Figure \ref{onenodeslide}, to take a $\Lambda_{(a,b)}$ curve to $\Lambda_{(a,b \pm a)}$ or to $\Lambda_{(a\pm b,b)}$. Using the Euclidean algorithm, with an appropriate choice of 1-handle slides, one can start with $T^*T^2 \cup \Lambda_{(a,b)}$, for any pair $a,b\in \mathbb{Z}$ that are relatively prime, and end with $T^*T^2\cup \Lambda_{(1,0)}$. That the Weinstein domains are symplectomorphic can also be proved using toric arguments (see \cite[Proposition 5.5]{ACGMMSW}).
\end{proof}

The diagram where we attach along the $(1,0)$ curve is preferable to the one where we attach along slope $(1,-1)$ (or a more complicated $(a,b)$ curve) because it is easier to simplify. Starting with Figure \ref{onenode}, we can perform Reidemeister moves, Gompf moves, handle cancellations and handle slides. We choose such a simplifying sequence in Figure \ref{onenodeseries} to obtain the diagram illustrated in Figure \ref{onenodefinal}.

\vskip.1cm

\section{A more complicated example: smoothing a toric divisor in $\mathbb{C}\P^2 \# 3\overline{\C\P}\,\!^2$}

Consider $\C\P^2$ blown up three times with the same size of the blow ups. The corresponding Delzant polytope is a hexagon illustrated in Figure \ref{hextor}. Observe that the sizes of the blow-ups have been chosen precisely to make the polytope centered with respect to all of its vertices. The inward normals of the six corners are: $(1,0),( 0,-1),(-1,-1),(-1,0),(0,1), (1,1)$. By Theorem~\ref{thm:toricWein}, if we smooth all six singularities that map under the moment map to the vertices of the hexagon polytope, the complement of the smoothed divisor is $T^*T^2$ with $2$-handles attached along
 $$\Lambda_{(1,0)},\Lambda_{(0,-1)}, \Lambda_{(-1,-1)},\Lambda_{(-1,0)}, \Lambda_{(0,1)},\Lambda_{(1,1)}$$
as shown in Figure \ref{hex}.

\begin{figure}[!ht]
  \centering
  \includegraphics[width=6cm]{figures/hextor.pdf}
  \caption{The Delzant polytope of  monotone $\C\P^2 \# 3\overline{\C\P}\,\!^2$.}
  \label{hextor}
\end{figure}

\begin{figure}[ht]
	\begin{center}
		\includegraphics[width=10cm]{figures/hexleg.pdf}
		\caption{The curves $(1,0),( 0,-1),(-1,-1),(-1,0),(0,1), (1,1)$ on $T^2$ before and after isotoping them to agree with concentric circles co-oriented inward (except where they pass through the $1$-handles).}
		\label{hex}
	\end{center}
\end{figure}

Figure \ref{hex} shows the curves $\Gamma=(1,0),( 0,-1),(-1,-1),(-1,0),(0,1), (1,1)$ on $T^2$. As in the previous section, we isotope the curves giving a Legendrian isotopy of their co-normal lifts, in order to have them run parallel to the boundary of the $0$-handle in $T^2$ except possibly at $1$-handles. This allows us to identify them in $J^1(S^1)$. We isotope each curve so that at the end, the co-orientation points inward. Since there are multiple curves, we choose an isotopy which minimizes the number of crossings.

\begin{figure}[ht]
	\begin{center}
		\includegraphics[width=10cm]{figures/hexjet.pdf}
		\caption{The curves in $J^1(S^1)$ corresponding to the curves $(1,0),( 0,-1),(-1,-1),(-1,0),(0,1), (1,1)$.}
		\label{hexjet}
	\end{center}
\end{figure}

Once these curves agree (except where they enter the $1$-handles) with parallel copies of suitable concentric circles which are positive Reeb push-offs of boundary of the $0$-handle of the $T^2$ $0$-section, we identify a neighborhood of boundary of the square (containing all our isotoped curves) with a neighborhood of the zero-section of $J^1(S^1)$. This identifies all our curves in $T^2$ with curves in $J^1(S^1)$ as in Figure \ref{hexjet}. We then satellite the image of the curves in $J^1(S^1)$ onto the image of $S^1$ in the Gompf diagram of $T^*T^2$, using the conventions described in the previous section and illustrated in Figures~\ref{fig:gradient} and~\ref{curvecorner} to maintain the relative positions of the curves. The result is Figure~\ref{hexleg}.

\begin{figure}[ht]
	\begin{center}
		\includegraphics[width=8cm]{figures/hexstart.pdf}
		\caption{A Legendrian handle diagram obtained by simplifying the Legendrian handlbody diagram of the complement of the toric divisor of $\C\P^2\#3\overline{\C\P^2}$ smoothed in six nodes.}
		\label{hexleg}
	\end{center}
\end{figure}

\begin{figure}[ht]
	\begin{center}
		\includegraphics[width=8 cm]{figures/hexfinal.pdf}
		\caption{A Legendrian link obtained by simplifying the Legendrian handle diagram of the complement of the toric divisor of $\C\P^2\#3\overline{\C\P^2}$ smoothed in six nodes.}
		\label{hexfinal}
	\end{center}
\end{figure}

Performing a sequence of handle slides and Legendrian isotopy yields a simpler Weinstein handle diagram shown in Figure \ref{hexfinal}. The Weinstein handle diagram allows us to easily compute the homology of $\C\P^2 \# 3 \overline{\C\P^2}\setminus\nu(\widetilde{\Sigma_1})$, where $\widetilde{\Sigma_1}$ is the smoothed toric divisor and $\nu(\widetilde{\Sigma_1})$ is the neighborhood of $\widetilde{\Sigma_1}$. The handle structure determines the cellular/Morse homology chain complex, which allows us to determine the homology groups. A presentation for the fundamental group can also be computed from the handle decomposition where the $1$-handles correspond to generators and $2$-handles provide relations.

In this example we obtain:
\begin{align*}
\pi_1(\C\P^2 \# 3 \overline{\C\P^2}\setminus\nu(\widetilde{\Sigma_1}); \mathbb{Z}) &= 0, \\
H_0(\C\P^2 \# 3 \overline{\C\P^2}\setminus\nu(\widetilde{\Sigma_1}); \mathbb{Z}) &= \mathbb{Z}, \\
H_1(\C\P^2 \# 3 \overline{\C\P^2}\setminus\nu(\widetilde{\Sigma_1}); \mathbb{Z}) &= 0, \\
H_2(\C\P^2 \# 3 \overline{\C\P^2}\setminus\nu(\widetilde{\Sigma_1}); \mathbb{Z}) &= \mathbb{Z}^{\oplus 5} \text{ with intersection form }
  \left[ {\begin{array}{ccccc}
   0 & -2 & 1 & 0 & 0\\
   -2 & -4 & -1 & 1 & 1\\
   1 & -1 & -2 & 1 & 0\\
   0 & 1 & 1 & -2 & -1\\
   0 & 1 & 0 & -1 & -3\\
  \end{array} } \right]
,\\
H_i(\C\P^2 \# 3 \overline{\C\P^2}\setminus\nu(\widetilde{\Sigma_1}); \mathbb{Z}) &=0 \text{ for $i=3,4$.}
\end{align*}

Using the same toric diagram, instead of smoothing all six nodes, we can choose to smooth only a subset of them. The symplectic topology of the corresponding complement will differ (with fewer handles attached). We consider here an example yielding a Weinstein manifold of interest.
 Let us smooth three alternating nodes in $\C\P^2 \# 3 \overline{\C\P^2}$. The complement of the smoothed divisor given by  $T^*T^2$ with $2$-handles attached along $\Lambda_{(1,0)}, \Lambda_{(-1,-1)}, \Lambda_{(0,1)}$ is pictured in Figure \ref{althexstart}.
 Performing a sequence of handle slides and Legendrian isotopy on Figure \ref{althexstart} yields a simpler Weinstein handle diagram shown in Figure~\ref{althexfinal} consisting of a Legendrian (2,4) torus knot.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=8 cm]{figures/althex.pdf}
\caption{A Legendrian handle diagram obtained by simplifying the Legendrian handlbody diagram of the complement of the toric divisor of $\C\P^2\#3\overline{\C\P^2}$ smoothed in three nodes.}
\label{althexstart}
\end{center}
\end{figure}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=5 cm]{figures/althexfinal.pdf}
\caption{A Legendrian (2,4)-torus link obtained from Figure \ref{althexstart} by a sequence of handleslides and Legendrian isotopy.}
\label{althexfinal}
\end{center}
\end{figure}

In this picture we can see two Lagrangian spheres in distinct homology classes in the complement of this smoothed divisor. Again, the homology of $\C\P^2 \# 3 \overline{\C\P^2}\setminus\nu(\widetilde{\Sigma_2})$, where $\widetilde{\Sigma_2}$ is the smoothed toric divisor and $\nu(\widetilde{\Sigma_2})$ is the neighborhood of $\widetilde{\Sigma_2}$ can be computed from the Weinstein handle diagram. In particular,
\begin{align*}
\pi_1(\C\P^2 \# 3 \overline{\C\P^2}\setminus\nu(\widetilde{\Sigma_2}); \mathbb{Z}) &= 0, \\
H_0(\C\P^2 \# 3 \overline{\C\P^2}\setminus\nu(\widetilde{\Sigma_2}); \mathbb{Z}) &= \mathbb{Z}, \\
H_1(\C\P^2 \# 3 \overline{\C\P^2}\setminus\nu(\widetilde{\Sigma_2}); \mathbb{Z}) &= 0, \\
H_2(\C\P^2 \# 3 \overline{\C\P^2}\setminus\nu(\widetilde{\Sigma_2}); \mathbb{Z}) &= \mathbb{Z}\oplus\mathbb{Z} \text{ with intersection form } \left[ {\begin{array}{cc}
   -2 & 2 \\
   2 & -2 \\
  \end{array} } \right]
,\\
H_i(\C\P^2 \# 3 \overline{\C\P^2}\setminus\nu(\widetilde{\Sigma_2}); \mathbb{Z}) &=0 \text{ for $i=3,4$.}
\end{align*}

\begin{figure}[h]
	\begin{center}
		\begin{tikzpicture}
			\node[inner sep=0] at (0,0) {\includegraphics[width=5 cm]{figures/althexlag.pdf}};
			\node at (-1.8,0){$x$};
			\node at (-0.9,0){$y$};
			\node at (0,0){$z$};
			\node at (1,0){$w$};
			\node at (0.9,0.9){$a$};
			\node at (0.9,-0.9){$b$};
			\node at (2.47,0.85){\textbullet};
			\node at (2.47,-0.9){\textbullet};
			\node at (2.8,0.85){$t_1$};
			\node at (2.8,-0.9){$t_2$};
		\end{tikzpicture}
		\caption{A diagram of the complement of the toric divisor of $\C\P^2\#3\overline{\C\P^2}$ smoothed in three nodes in the Lagrangian projection with labelled Reeb chords and marked points generating the Chekanov-Eliashberg DGA.}
		\label{althexlag}
	\end{center}
\end{figure}

The Weinstein diagram allows us to also compute the DGA of $\Lambda \in S^3$, $\mathcal{A}(\Lambda)$ (see~\cite{NE} for a good survey on how to calculate the Legendrian DGA). In this case is generated by $a,b,x,y,z,w$ and marked points $t_1$ and $t_2$, as shown in Figure \ref{althexlag}. The generators have gradings
$$|a|=|b|=1, |x|=|y|=|z|=|w|=0$$
with differential
\begin{align*}
\partial x &= \partial y=\partial z =\partial w=0\\
\partial a   &= xyzw+xw+xy+zw+1+t_1^{-1}\\
\partial ab  &= wzyx+yx+wz+wx+1+t_2.
\end{align*}

The DGA of $\Lambda$ is an interesting invariant to consider because the Wrapped Fukaya chain complex of $\C\P^2 \# 3 \overline{\C\P^2}\setminus\nu(\widetilde{\Sigma_2})$ is $A_{\infty}$-quasi isomorphic to the $DGA$ $\mathcal{A}(\Lambda)$~\cite{BEE,Ekholm,EkholmLekili}. Note that $\mathcal{A}(\Lambda)$ has no negatively graded Reeb chords and therefore the degree-$0$ Legendrian contact homology of $\mathcal{A}(\Lambda)$, denoted by $LCH_0(\Lambda)$, is finitely generated. Furthermore, it is expected that $LCH_0(\Lambda)$ is Morita equivalent to a commutative ring $R$ such that $\hat{X}=Spec (R)$ where $\hat{X}$ is the mirror of $\C\P^2 \# 3 \overline{\C\P^2}\setminus\nu(\widetilde{\Sigma_2})$. See~\cite{CM, EtguLekili} for other examples of such a phenomenon.

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text:
Q: PHP mysql: 2 similar queries echoes results of 2nd query only The result of the 2nd query overwrites the result of the 1st query.
I've never run into a similar issue before.
mysql table:
updateid    picture  movie
  14          1       1
  22          0       1
  33          1       0

php file:

<?php
$image_check= query("SELECT picture FROM list WHERE updateid = '$updateid' ");
foreach ($row as $image_check);

if (!($image_check))
{
?>

    <div class="result" id="camera<?php echo $updateid ?>">FOO

<?php
}
else
{
    ?>
    <div class="result" id="camera<?php echo $updateid ?>">BAR

    <?php
}

?>

  <?php
  $movie_check = query("SELECT movie FROM list WHERE updateid = '$updateid' ");
  foreach ($row as $movie_check);
    if  (!($movie_check))
    {
    ?>
 <div class="result1" id="movie<?php echo $updateid ?>">FOO

<?php
}
else
{
    ?>
    <div class="result1" id="movie<?php echo $updateid ?>">BAR

    <?php
}

?>

when i run those 2 queries, the result for
14 echoes BAR BAR
22 echoes FOO FOO
33 echoes FOO FOO
Any ideas what might go wrong here ?

A: Probably typos. you're using variable variables:
  foreach ($row as $$movie_check);
                   ^^--- note the doubled $

e.g.
$x = 'foo';
$foo = 'bar';
echo $$x; // outputs 'bar'
echo $x; // outputs 'foo'

As a general rule, never EVER use variable variables. They make for utterly unmaintainable and nearly impossible-to-debug code.

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Simon Cowell and his longtime partner, Lauren Silverman, are getting married, a representative for the "America's Got Talent" judge confirmed to CNN on Wednesday.The representative gave no further details, but the Sun reported that the pair got engaged during a recent break to Barbados.Cowell, aged 62, and Silverman, now 44, began their public relationship in 2013, shortly after Silverman's ex-husband and Cowell's close friend, Andrew Silverman, filed for divorce in July, claiming the pair were involved in an extramarital affair."The X Factor" creator and Silverman, a New York socialite, welcomed their first son, Eric, in 2014. Silverman also has a son from a previous marriage.When previously asked on fellow "American Idol" alum Ryan Seacrest's radio show if he planned to marry Silverman, Cowell said, "Let's put it this way, I wouldn't expect her to be a single mother."The reality TV executive told Britain's OK! magazine in November that he and Silverman are "closer than ever," revealing the "romance is still alive" between them."COVID-19 was the real test," he said."Like everyone, we were in lockdown for a long time and that's when you realize whether or not you actually enjoy each other's company or not. And we really, really did. So to answer your question... yes, the romance is still alive!"
Simon Cowell and his longtime partner, Lauren Silverman, are getting married, a representative for the "America's Got Talent" judge confirmed to CNN on Wednesday.
The representative gave no further details, but the Sun reported that the pair got engaged during a recent break to Barbados.
Cowell, aged 62, and Silverman, now 44, began their public relationship in 2013, shortly after Silverman's ex-husband and Cowell's close friend, Andrew Silverman, filed for divorce in July, claiming the pair were involved in an extramarital affair.
"The X Factor" creator and Silverman, a New York socialite, welcomed their first son, Eric, in 2014. Silverman also has a son from a previous marriage.
When previously asked on fellow "American Idol" alum Ryan Seacrest's radio show if he planned to marry Silverman, Cowell said, "Let's put it this way, I wouldn't expect her to be a single mother."
The reality TV executive told Britain's OK! magazine in November that he and Silverman are "closer than ever," revealing the "romance is still alive" between them.
"COVID-19 was the real test," he said.
"Like everyone, we were in lockdown for a long time and that's when you realize whether or not you actually enjoy each other's company or not. And we really, really did. So to answer your question... yes, the romance is still alive!"


text:
package de.blogspot.mszalbach.iss.domain;

import com.fasterxml.jackson.annotation.JsonIgnore;
import com.fasterxml.jackson.annotation.JsonIgnoreProperties;
import com.fasterxml.jackson.annotation.JsonProperty;
import de.blogspot.mszalbach.iss.validator.ISIN;
import org.hibernate.envers.Audited;
import org.squirrelframework.foundation.fsm.StateMachineData;

import javax.persistence.Column;
import javax.persistence.Entity;
import javax.persistence.GeneratedValue;
import javax.persistence.Id;
import javax.validation.constraints.DecimalMin;
import java.math.BigDecimal;

/**
 * Security Entity.
 */
@Entity
@Audited
@JsonIgnoreProperties(ignoreUnknown = true)
public class Security {

    @Id
    @GeneratedValue
    private Long id;
    @ISIN
    @JsonProperty(required = true)
    private String isin;
    private String symbol;
    @Column(scale = 2)
    @DecimalMin("0")
    private BigDecimal nominalValue;
    private String issuer;
    @JsonProperty(access = JsonProperty.Access.READ_ONLY)
    private String state;

    @JsonIgnore
    @Column(columnDefinition = "TEXT")
    private StateMachineData.Reader stateMachine;

    public Security() {
    }

    public Security(String isin, String symbol) {
        this.isin = isin;
        this.symbol = symbol;
    }

    public String getState() {
        return state;
    }

    public Long getId() {
        return id;
    }

    public String getIsin() {
        return isin;
    }

    public void setIsin(String isin) {
        this.isin = isin;
    }

    public String getSymbol() {
        return symbol;
    }

    public void setSymbol(String symbol) {
        this.symbol = symbol;
    }

    public BigDecimal getNominalValue() {
        return nominalValue;
    }

    public void setNominalValue(BigDecimal nominalValue) {
        this.nominalValue = nominalValue;
    }

    public String getIssuer() {
        return issuer;
    }

    public void setIssuer(String issuer) {
        this.issuer = issuer;
    }

    public StateMachineData.Reader getStateMachine() {
        return this.stateMachine;
    }

    public void setStateMachine(StateMachineData.Reader stateMachine) {
        this.state = stateMachine.currentState().toString();
        this.stateMachine = stateMachine;
    }

}

redpajama_set_name: RedPajamaGithub


text:
Q: getting strange character while loading view in codeigniter Getting strange character while loading a view in codeigniter view.
Main View
<!DOCTYPE html>
<html>
<head>
<?PHP $this->load->view("common/header"); ?>
</head>
<body class="page-header-fixed">
</body>
</html>

Header View
<title><?PHP echo $this->config->item('project_name'); ?>| Dashboard</title>
<meta content="width=device-width, initial-scale=1" name="viewport"/>
<meta charset="UTF-8">
<meta name="description" content="<?PHP echo $this->config->item('project_name'); ?> | Dashboard" />
<meta name="keywords" content="admin,dashboard" />

when i inspect in chrome it's shows  above of loaded view and put space in view
please see attached image

A: The character in question &#65279 is the Unicode Character 'ZERO WIDTH NO-BREAK SPACE' (U+FEFF). It may be that you copied it into your code via a copy/paste without realizing it. The fact that it's not visible makes it hard to tell if you're using an editor that displays actual unicode characters.
Why is  appearing in my HTML?


Between 1918 and 1921 the U.S. Shipping Board established several routes and services and one of these was the American Colombian Line, managed by the Clyde Steamship Co. This led to a trade war with United Fruit Co. and it was agreed that the Caribbean trade of Clyde SS Co. would be separated from their domestic coastal trade. The Colombian Steamship Co., New York was then formed to operate a route from New York to Puerto Colombia, Cartagena and later to Haiti. The service expanded to the Leeward and Windward Islands and later to Brazil and other U.S. East Coast ports. In 1938 Colombian Line was consolidated with Grace Line which led to the extinction of Colombian Line as a seperate entity.
Jamaica see United Fruit Co. Peten. .
Pastores see United Fruit Co. Pastores. .


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