My 2018 season was really the best season that I've ever had mentally and physically. It was a lot of things that I was going through mentally, to be honest, and I had a lot of people behind my back to be able to help me and pull me through, especially with the Seahawk staff and stuff like that, and everything they did for me was amazing. Now when we get to the football side of it, I did everything that I knew that I could possibly do and that I could bring to the table. I understood that it was all about being efficient. It wasn't about how many targets that I got. If I got 150 targets, I wanted to make sure I had caught the ball 150 times, or I caught every pass that was capable of me being able to catch the ball. I think for me, that was the biggest thing that I wanted to focus on and do. I wanted to be efficient.  For me, it's all about what the team wants me to be able to do. A lot of people thought that I weren't able to accomplish some of the things that I accomplished, but the biggest thing is when you're a team player, you're probably never going to accomplish the things in life that a lot of people have expectations for you to be able to do. Obviously, I mean, maybe it'd be different if I had 150 targets. Who knows what would have happened? I could have still had a way better season, 2,000 yards, or I could have had 500 yards, and people said, why are we throwing them the ball that much? But for me, it was all about what can I do to make this team better? What can I do to be able to bring something to the table that would be special to everybody around me? And all I want to do is be able to make a difference, not only in my play, but also in everything that I do in life as well as I'm on this team. You got to be able to count the little things. Like the little things in life is what makes everything like a monumental moment, and that's exactly what it was for me. It's going back to the little things, simplifying life and what it really means for me to be able to live and be happy. To me, ever since I've been there, he's been a big brother to me. And like that's something that I say. I don't really say people are big brothers to me a lot, but he's done nothing but try to help me be successful and help me be the best that I could be in life. Some people might think it's corny, but like I say, sometimes you need a corny person in your life. Like you might need somebody to be able to remind you like, hey, let's get it going. Like, hey, come on, we got this. Like sometimes we get tired of hearing that, but when you don't have that, imagine if we ain't had that in our locker room. Ain't no telling what would happen. Honestly, if I had to give a prediction of what I think is going to happen, I would say this. If it's a shootout, the Rams will win the game. If the Rams get on top early and we, and they have to make time, Brady throw the ball to be able to win the game. I think the Rams win the game. Not because I don't think Tom Brady can do it. He could do anything. I just think the pass rush is what makes them so special, what makes the Rams so special. If it turns out to be a low scoring game, or even if it's a high scoring game, but the Rams offense cannot play, if the Rams offense doesn't play good, Tom Brady wins. The Patriots win. Super Bowl MVP, I would say if it was for the Rams, I would say if it doesn't go to the quarterback golf, it goes to Robert Woods. I feel like he's a sleeper that will make the difference, that will change everything for him. Offensively, if I had to pick for the Patriots, it would be Tom Brady. I was blessed to be able to partner with Pisa Hut this year, and for me, I'm very thankful for that. Reason because that's something that I used to eat when I was younger, and the other thing about it is, like when you grow up and you play football, you never really know or think that you could be able to represent companies like this. So having the chance to be able to represent this type of company, I got to see some of the things that they want to be able to do. One of the things that we did that I thought was special was the booking program, where we teach kids how to be able to read, and we try to show them, like, hey, knowledge is power.  But it ended up working out perfectly. I got to do it with a Seagal girl, a Seagal girl, one of the cheerleaders for the Seahawks, and it was amazing. And also the last thing that I want to be able to tell you guys is this. If you purchase pizza or wings on Super Bowl Sunday, you get 20% off at the NFL shop. So you know what that means. Go buy Tyler Lockett, Jersey. Just letting y'all know that right now. Tyler Lockett, number 16, Seattle Seahawks.


We are Siamese if you please. Is that even politically correct anymore? No. No, I don't think it was politically correct then. Yeah. It was very racist.  Yeah. You ever see Peter Pan? There's a song called What Makes the Red Man Red? They're not even trying to hide it. What Makes the Red Man Red? What Makes the Red Man Red?  You did some Corbin. I can fly. Also, a song of the South. Yeah. They don't even have that one on Disney Plus. Which is interesting. They have that ride at Disneyland, but I think it's going away now. It's definitely going away. Isn't it? Yeah. It's a shame because Disney tries to bury that. Yeah, they really do. You can't find song of the South anywhere because you know the song, the... That's about a slave just being happy being a slave. That's Uncle Remus. Not good. The Tales of Song of the South and the Tales of Uncle Remus who was, he's a slave who's happy to be a slave. Yeah. Anyway. Not great. Anyway. I'm on this tangent. But today, what are we doing, Rick? We're going to squeeze the milk out of cows. Yeah, we are. Today, it's the top 10 most beautiful train journeys in India. Oh, cool. One of the things we didn't get to do, sadly, even though we had a very short trip and a very jam-packed trip. Next time, we'd love to take a train, like a train ride on. We're from High Jacket Train. Yeah, from one place that we are. Just like Jesse James. Yeah. To travel to wherever we're going next because we know it's such a common thing and also they have really cool trains as well. Yeah. You've seen several videos with train stuff. Yeah. But this is the most beautiful and I'm sure we're going to see it. The train in Ireland? No, because I was only there four days. Okay. I was trying to get everywhere fast as I could. Yeah. I think trains and subways are like, if you can and you've got the time, in certain cities especially, like if you go to New York, ride the subway. Yeah. It's just a cool experience. Here we go. Namaskar, friends. How are you? Friends. Once again, welcome to your channel, Top 10 Bird. Like a game show. Yeah. Woo. Great shot. Good job narrating. We're having happiness and energy. Haha. Yeah. Woo. Woo. Yeah. Woo. Woo. Woo. Go after it. Woo. Woo. Woo. Woo. Woo. Woo. Woo. Woo. Woo. Woo. Woo. Woo. Woo. Woo. Konkan Railway. This rail was made to end the distance between Mumbai and Mangalore. Like a guard? It was kept by Eshwari Dharan Metro. This was the most difficult rail path in India. Konkan Railway, three states, Karnataka, Goa and Maharashtra. We can take this one next time. Yeah, we can. Mountains, mountains, rivers, rivers, flowers. Absolutely, it should be fantastic. Oh, that's... You can't see that shot from the trek. It's a slow train and you can watch flowers do that. As a result, Neel Giri Express is a low-prey and only one-wheeled train that runs on these Indian railways. It has a distance of up to 26 km from Mettupalliam to Uddhugam Mandalam Puti. This train has 208 curves and 250 full parts. From the beginning, it takes 408 hours to reach Gantavwada. From 1908, Neel Giri Passenger still runs from Mettupalliam to Uddhugam Mandalam Puti. That's cool. Because the trains Neel Giri, Mountain, Moti Pine, Oak and Neel Giri's jungles, Jhukav, Vagra and Srinagar are all here. This is the 13th track with the highest number of trains. It has Americans built on the railroads. And then they just kind of... Neel Giri has the audio. From Mottalliam to Uddhugam, Neel Giri Express, Neel Giri Express, Grand Hulak, Adriely, Hilgrove, Kateri, Rani Mede, Kateri, Kunur and the station like Love Devil. It travels from Mottalliam to Dhalan for about 5 hours to the distance of 26 km. Do you know that the movie's title is Chaniya Chaniya. It was shot here. Chaniya Chaniya Chaniya Chaniya. It is the distance of 407 km from Tamil Nadu to Rameshwaram. which takes a lot of time to decide, and in that time, you take the path of the road between them. It is also known as Mahazagar's special travel. From Pambandpul to Rameswaram, apart from the train Chhavisroth Romance and Romance, in Tamil Nadu, you can travel from Mandak to Pambandweep, to Rameswaram Shah and Shanti Rukpan. This is one of the most beautiful Indian train travels in India. One of the most beautiful railways in India, this is one of India's most long-distance, Palat Strait, which is one of the main roads that connects the main land India to Pambandweep. 4. Kajrat-Lonabla It is 28 km away from Kajrat. It passes through an expensive place in the form of a train. This is carried out in the areas of Udaar-Pashtibi Ghatu, which are known as the Jai Vibhidha. During the rainy season, this rail path becomes much more beautiful along with the winding roads and many potholes. Along with the mountainous terrain, a beautiful view is made from the window of your train. Before the journey of Bandar Hill, through the train Thakurwadi village, it passes through Criss-Cross in Khandala. As soon as Kajrat rail starts, it reaches Lonabla through Khandala. 5. Toy Train Darjeeling Himalaya Railway New Jalpai Gudi Darjeeling One of the oldest railways of Darjeeling Himalaya Railway India, which was built around 1880. This journey will be done through the railway, which is two-foot Sankirna Gesh and is based on the Roopline Krat Yogi and Rish Zag. This railway is 28 km away from New Jalpai Gudi. It ends in Darjeeling along with 12 trains all the time. Trains are built in the morning. In India, the oldest railway in the world, the Toy Train is also a western railway station, which was announced by UNESCO in 1999. Railway Shaqha Loops is one of the special places of this railway. The most popular is the Batasya Loop, which offers a magnificent view. With Darjeeling and Barf, you can see the beauty of Mount Kancha Junga. You will definitely want to enjoy this train journey along with some darts and jeans. That's a brand that's very well-known here. I don't think anybody in India loves it. If you want to enjoy the roads, the roads and the fields of Dhundalee Ghans, then you should travel in the train. I just want to do that. There would never be a lot to happen in America. Too many losses. That's what I would do though. Now you can go on a bus like that. You're just a lot of... Oh man. There are some trains on this route. This journey is really fun, but slowly the train goes to reach the capital of summer, Dharampur, Solan, Kanda Ghat, Tara Devi, Salokra, and through summer health stations, it goes from Lai Baddharoop to Dhalan. Kaalka Shimla Railway, 2.6 feet narrow gauge railway, covers the distance between Kaalka and Shimla. This is one of the most dangerous railway roads in the world. This is one of the best Indian trains in the world. Saad, Nilambur, Shurnur. Shurnur Railway Line is one of the most broad gauge railway lines in the Lakhkar Division of Nilambudh Dakshan Railway. Kerala? Kerala? Kerala? Kerala? Kerala? Kerala? Kerala? Kerala? Kerala? Kerala? Kerala? Kerala? Kerala? Krishnan才imeshaayya... I am very happy to be here. Like the famous Pariyatak Akharshan near Neelambur, the longest saagon tree in the world, Adyan Parajal Falls, Ganun plot, Ullattaka, Neid Mumayam, Banglaa Kunnu, Koi Parajharni, Bansura Mandir, Vanyambalam. 8. Kashmir Railway, Jammu Uthampur Kashmir Railway is one of the most beautiful railway lines in India, which passes from Pishurango and Ulamnagar for hundreds of years, which is the highest railway bridge in India, which is also known as Nirmana Dheem. Kashmir Railway Line is the most challenging railway in the Indian Railway industry, which crosses the highest mountain range and areas of Bukampuk, and in this journey, you will find the beautiful Shivangat Veyn, which you will feel like you are traveling in heaven. 9. Rajasthan Rani, Jodhpur, Jaisalmer. In Jodhpur, Jaisalmer, it is about 285 km away. Rajasthan Rani is one of the most famous tourist places in India. This is the most famous railway in Rajasthan. We have seen that. I think we have seen that. The Royal Railway, yeah, I think we have seen that. Oh! Rajasthan is on a different level. Oh!  They serve the guests of the train here. Yes. This is a train from Arawali to India. So tell me, is it fun? 10. Koraput Rai Gada, Koraput Rai Gada Line, or KKR Line is a branch of Kirtanbhul-Kothaabals Railway Line. Koraput, Andhera and Ghani Jungle, Meadows, Pani Girney and Teres Wali Ghatis pass through. On the land of Rai Gadakshetra Aadivasi Odisha, it is a beautiful and beautiful place to visit. Which is a heaven. While Rai Gada is 200 meters above the sea level. Koraput is about 800 meters high. When the train goes from Koraput to Rai Gada, the view of the downhill looks beautiful. The shape of the horse's shoes is combined with the shape of the mountains. So friends, this is the top 10 railways in India that you will definitely want to visit. Don't forget to watch our Shri Yash Siddhas Chanda and Savari video. Yeah. No, I wish, I wish America were planning on building the railways. And specifically, I think it was Ford and a couple of the car companies. We had this big plan to have a big metro thing just like New York. Correct. In the early 1900s, it was the car industry. The car industry bought it out because they thought it would get into their profits. Yeah. And now we have the worst traffic. Yeah. And America was built on the railway system. I mean, the expansion from east to west and boom towns that popped up along the railway systems and the expansion of the west were what made America when we expanded. And yes, it was the automobile and the auto industry that just really... We still have them, but they're not. No. The industry, it's called the Amtrak. Yeah. I mean, it's nice. But it's just one train system, basically. Pretty much, unfortunately. There's no metro rails here, but it's... So maybe one day. But yeah, I've... Obviously, we've heard a bunch about India's trains and all the different stuff that places you can go, but I'd go on every one of these. Oh yeah, 100%. Because obviously, what we know of the Indian topography, just it would be just gorgeous to look at. Just like when I was in Ireland, it would be gorgeous to go on trains. It's a great way to see the land and a lot of the land in a fast amount of time, but slow enough to be really cool to make that a trip. Like let's go to India and do those 10 trains. It would be amazing. Next time we go to India whenever COVID allows in the year 2023. Who knows? Hopefully not. But you know, it's bleak right now. But whenever we do, like maybe that one from... Because we're probably going to start in Mumbai and then we're going to go south. Go south, basically. You know, Kerala, Goa, Tamil Nadu, all those places down south. But there's really one beautiful train from Mumbai. Take that down. You can go from Tagoa to somewhere else after that. Sounds good to me. Might be a good option. I'm sure a lot of you know the good trains to go on. Obviously, whenever we actually plan it out, we'll be able to see better. Trains are something that you have to have time for. Exactly. We've got the plan. We jam-packed our trip. We shouldn't have been able to do all we did in the short amount of time. We were exhausted. Well, our poor people that were hosting us. I felt so bad sometimes because they just... They wanted to hang out and hang out. They were so gracious, especially because we spent most of our time on the trip in Mumbai. And that was toward the end of the trip. But so often our hearts were to spend time with them, but we were fried. And they just were gracious. Yeah, because we even canceled something with them. Straight up canceled something because we were just too exhausted. Yeah, because we get there. We're already jet lagged from our 22-hour plane flight.  I think we had five or six hours of sleep the first night. Right? Yeah. We got up. We had South Indian breakfast in Delhi. And went straight to a meet-and-greet that day. And you all loved it. Yeah, you all. That was your favorite video of the trip. Although I will say, if you want to do something really fun, do yourself a favor. My first trip to India was when I didn't get my e-visa to properly enter the country. And I was supposed to spend a six-day trip in Calcutta and it turned into a three-day trip. Yeah, so fly, do a flight into India. And get adjusted to the time. You had like a 37-hour flight. And fly back. Yeah, I did. Because I had like four planes. Four planes. Yeah, I had a long way over. Exactly. I made sure we had very short. It was beautiful. It was much more lovely. But let us know what strain ride, whenever we do come back, COVID permitting. Because I don't think we're still allowed in India. Next year. I don't think any country has lifted their restrictions for America. Last I checked, we were still only allowed into 22 countries in the world. Yeah. And I would be surprised if it shrunk from there. Yeah, it's shrinking because they're exploding all over right now. Yeah, everybody's exploding. Like, there's no reason anybody should be allowing anybody in right now. We're gonna see a baby boom in a couple years. Because people are at home with nothing to do. In nine months. Yeah. Well, actually some of them already might be coming out. Coming out now. COVID babies. Anyways, let us know what strain we should take next time we come down below.� two mothers child.


and welcome to, actually, let's just skip the intro. You guys already know what it is. My hands are tired from, you know, keep on rubbing for 83 episodes. And you know, I'm tired from saying, look at the clock behind me, it's 10.30. You guys know what it is. It's time for hashtag LNT, episode 83.  I had to look to remember the word, but yes, we are celebrating tonight. And what a celebration it is. The guys actually, you know, Masha'Allah, put in effort and some time to decorate the studio. So shout out to Bashir Assadi and all of them who, and Mustafa Nazir and, of course, Ali Meitham. It is the biggest day of the year. So I'd like to give a quick shout out to everyone participating in tonight's episode from Hassan Asadid Salmad al-Hussaini to Hassan Hadi, a.k.a. Sota V, to Ali Jassim. And of course, I gotta give a shout out to myself as well. Ali Meitham, I already gave him a shout out, a.k.a. Al-Lush Al-Hazine. Yes, we are celebrating tonight, but let's go take a quick break, come back, talk about what's ready in a few seconds. Once again, once again, we welcome everyone who's joining us tonight. As I mentioned, the studio looks fresh. I look fresh. The crew looks fresh. Everyone's having a good time tonight. And we're trying to kick it out, or kick it out with what's trending. Now, if you're infiltrated with history and historical antiquities, dating back to special wars in history, then look no further. If you're specifically in World War II, all you gotta do is go to Jersey, containing Radio Tower, that was previously a former Second World War fortress, and is now a vacation rental. Full screen, that looks amazing. One of the best views you can get. Now, this was built by the Nazis, Nazi forces in occupied Jersey during the Second World War. Those who actually go there, get to see a 360 view of both the land and the water together. So it's an amazing view. If you're thinking of a honeymoon, why not go for an antique from World War II? Your marriage can become very healthy after that. I'm just kidding. But no, really, if you guys need a job, I mean, try to go and apply it to become a pilot. Why? Because the Asian Pacific will now need the greatest number of pilots, technicians, and cabin crew for over the next two decades. How much do they need exactly? They need 240,000 pilots because their economic growth, it's so high, and it's skyrocketing. By 2037, they're gonna need 240 pilots, 317,000 cabin crew. That's a huge number, huge number. But if you guys don't know how to fly, just buy a kite, fly it, you know, in Iraq and probably find that during Ramadan especially, fly it, and then you know how to fly an airplane. I'm just kidding. Let's just jump into tonight's topic really quickly. Once again, we do welcome everyone joining us tonight if you're wondering why the lights, why everything is looking so fresh, why I look so fresh, give me a close up of myself on Aid Al Ghadir.  Tonight is something amazing. Although we do, sorry, sorry, sorry, sorry, sorry. Once again, we do welcome everyone joining us tonight. Now, as everyone knows, there are many wonderful and joyous celebrations around the world. Some cultures and religions commemorate each or a specific day and mark it as a day of celebration, giving gifts, celebrating in it, having family, over going to family, visiting parents, buying some sweets, looking fresh, you know, have some nice decorations, so on and so forth. Now for examples, Christians, they have Christmas and the Christmas sales are amazing, subhanallah. But for Christmas, I just, you know, do I really have to say what Christmas is? It's the birth of Jesus Christ. But if we move on to Judaism, the Jewish have Hanukkah festival of the lights where they celebrate the redundant of the dedication of the second temple later on in Jerusalem. But if we were to come to last, but not least, Islam. Now for all those watching us and everyone out there, we know that within Islam, there are several aides. The first is being aid al-Fatah. Aid al-Fatah means I say, you know, Ramadan is done.  And then after comes aid al-Fatah, which we just had a week ago. I know a lot of this sort about it, about a week ago. And then, you know, it's the festival of sacrifice. But there's one specific aid that really hasn't gotten has gotten a lot of media attention, but not to the extent where everybody knows about it as much as the other do. Although it's more important, but we're trying to find out tonight what that aid is.  We're trying to ask the question to you guys, and the question's popping up in three, two, one, why is aid al-Ghadeel the greatest aid? That's a very simple question. Why is aid al-Ghadeel the greatest aid? Very simple and informative. The number you call us is plus 964-774-067-1836 and let us know what you guys think via that question right there that we just showed you. You can call us via a WhatsApp message, a phone call, and you can show us a voice message regarding tonight's topic, very simple. Why is aid al-Ghadeel the greatest aid? Let's take a quick break. Come back to you guys very short. Now, we do remind everyone of the question what we're talking about tonight. We're talking about aid al-Ghadeel, the best celebration out there. The celebration where she, I just go wild, now I'm just kidding.   There's another aid, there's aid al-Ghadeel, yabra, another one. We have another aid called aid al-Ghadeel. Now, so what exactly is aid al-Ghadeel? Why do Shi'a and Muslims celebrate it? Now, it's a very, very important question to discuss tonight, although it has been discussed by many, but it's good to open up the discussion for tonight as we're about to dive deep into what aid al-Ghadeel represents. Now, today around the world, Shi'as, a.k.a. Ahmad Ali, they're celebrating, and hashtag in here, are celebrating aid al-Ghadeel, and there are various facts and various narrations regarding this very, very important aid. Now, it's where the Shi'as get to celebrate the day where the cousin of Rasulullah, not only that, but his son-in-law, where he becomes his successor, the Prophet's successor. An amazing day where over 100,000 individuals, present on one day, on one occasion, in one place, and the Prophet is addressing them. Now, when and where did this happen? Now, as you know, today is 18th of the al-Hajjah, or night of the 18th of the al-Hajjah. 10 years after Hijjala, after the Prophet's migration to Medina, the Prophet goes to Hajj. After his return from Hajj, there's one specific area where everyone, every tribe that comes, or whoever comes to celebrate, or sorry, of whoever goes to Hajj on their way back to their home, they have to pass through this area called Khom. There's a small pond in that area. It's like with hills and stuff. If you check out the picture and on the map, you can probably see, but in that area, it's hills, there's a pond called Ghadir, where Prophet stood and gave the sermon of Ghadir Khom. Modern day Juhfa in Saudi Arabia. Now, this area right there, Ghadir Khom. Now, this area specifically, as I mentioned, is the area where all the tribes, whoever went to Hajj on their way back, they pass through here. So it's like a transit for them. They pass through there, greet each other, and then everyone goes to their own separate ways. However, Prophet Muhammad chose this area for this significant, because it's significant. Everyone is just gathered there. Over 100,000 individuals on that specific day. And trust me when I say, 100,000 isn't something small. It's huge. Now, on the 18th of the Hajj, 10th year after Hijrah, the Prophet stops in that area and stops everyone. Over there in the front stops in the back, over there in the back stops in the front. Everyone stops and listens to what the Prophet is about to say. And, you know, let's just keep in mind, this is the first and last Hajj of Prophet Muhammad. His official Hajj, first and last Hajj to Mecca. So everyone stops in this area called Khom, Ghadir Khom. Now, as soon as they stop, or the reason why they stop, Allah Subhanahu Wa Ta'ala sends Jubra'il to Prophet Muhammad and he says this exact verse in Al-Ma'ida, verse 67. He says, oh, Apostle, deliver what has been sent to you from your Lord and if you did not, then you have not delivered his message and Allah will protect you from the people.  The Prophet going through miseries, 23 years just to preach Islam, Allah Subhanahu Wa Ta'ala comes on this day and tells Prophet Muhammad, doesn't come, but sends Jubra'il, tells Prophet Muhammad, oh, Apostle of God, deliver or convey what has been sent down to you from your Lord and if you do not convey it, then it's like you haven't done anything. These 23 years are meaningless if you don't convey this specific message. Prophet Muhammad was, you know, it's hard, you know, a lot of people, when they analyze this, you can probably find this in lectures or on articles online. They have tried to use the word threatened, God threatens Prophet Muhammad, absolutely not. But it's in a way where an individual, you know, for example, if you're trying to build a house and one specific area where the house is built on is not set right, then that whole house is broken apart. Yes, Prophet Muhammad, 23 years of hardships, 23 years of misery and trying to bring everyone together to 100,000 on Ghadir Khom, the Prophet is trying to tell everyone that at the peak of Islam, at that time, at that era, he wants to let everyone know of what happened or what is going to happen on this specific day and why it's so special. So realize how Allah SWT says this announcement, this specific message that he has, the last message that he has to convey to everyone. Now, the sermon of Prophet Muhammad, as everyone knows on this day is pretty long, it's pretty lengthy. But I don't wanna keep on reading the whole thing and letting you guys, you guys can probably go read that in various books. But tonight, I wanna focus on the part where it's most important where Prophet Muhammad announces this message. What is that message?  I am worthy of the believers than themselves, of whoms ever. Then he raised the hand of Ali Mubalib and he says, whoever I was their Mola, I was their guardian, I was their leader, then Ali is the exact same in that specific picture right there. Raises his hand, he says, oh, people, whoever I was the leader of, whoever I was the Mola, the guardian of, then Ali is the exact same, then Ali is his master, is his Mola. Oh, Allah, be the supporter of those who support him, aka Ali, and an enemy of those who oppose him, aka Ali Nabi Talib. So it's important to keep that in mind when we're talking about this. Now all the schools of thought, you know, if you were to analyze what actually happened on that day and who was present on that day, the great historians, great Islamic historians were present at that time. Those who have narrated the books are present on that time, historians present on that time. And as some say, historians approximately over 300 historians were present on that day. So 300 times that narration is mentioned, over the 300 times that narration is mentioned. Now according to Phukah, jurisprudence, according to the scale as to how you should rate a narration, Mu'tawater is something that everyone's agreed on. Everyone has agreed on that. They call it the Mu'tawater, something that a lot of people have mentioned and a lot of people have agreed upon. Ghadir Khom, the sermon, the event, everything about it is agreed on. However, the split comes when we're trying to interpret the word Wali. You know, see, hence we get Ali and Wali Allah, Wali, Ali, Wali Allah in the Adhan. A lot of people try to use the fact that you know what? Wali simply means a spiritual leader, a friend of Prophet Muhammad, blood related to Prophet Muhammad. Not really focusing on the fact that Wali, if you were to go actually interpret it, within Arabic, within Arabic, you'll find it meaning a successor. And plus, when Prophet Muhammad wants to say whoever I was his Mawla, what Mawla means blood related, so 100,000 people, the Prophet wasn't related to 100,000 people. He wasn't, he was a friend, but he wasn't related to 100,000 people. So whoever I was their Wali, then Ali is their Wali, or Subhan Allah, so whoever the Prophet is related to, then Ali is really that, that's common sense. When someone actually tries to come and give this kind of proof, and give this kind of evidence to such an argument, then they have to rethink of it. Rethink of what to say and how to say it. Now, this is where the split happens. A lot of people say that, you know what, Ali Al-Talib is a spiritual leader, and she as believed, the Twelvers, the Ethna, Ashuris, the Jafaris, they believe that no, Wali means that he is the successor, he is the leader, and he is the one that comes after Prophet Muhammad, righteously. Some things did happen, and history won't be kind, but yes, things happened, and Ali wasn't the successor, was wrong. We don't want to get into that, today is what we're trying to celebrate. So what we're trying to say is that the word Wali is what split the two being as a Sunni narration and as a Shi'a narration.  This is where they get their conclusion from, that the Prophet is saying, yes, support Ali, and they say we do support him, but he's not our Wali. It's way deeper than as we mentioned earlier on. Then if we were to go and continue the discussion, but up to now, everything should be clear of why Eid al-Ghadeer is the greatest Eid.  You know, it finishes a month of fasting, a month of obedience, a month of discipline, self-discipline, a month, while Allah, Subhanahu wa Ta'ala, says that within that month, there's layat al-Qadr, which is greater than a thousand months. And within that, the Quran was revealed. So at the end of that month, Eid comes, shouldn't that be the greatest Eid? Let's hold that thought for a second. The second Eid we have, we have Eid al-Adha, sacrifice, Prophet Abraham, and his trying, him trying to sacrifice, his son for Allah, Subhanahu wa Ta'ala, and then that ritual just passed down to Islam and how everyone now sacrifices on that day, they call it the Eid al-Adha. Isn't that the greatest Eid? Where an individual sacrifices to Allah, Subhanahu wa Ta'ala, goes to Hajj, one of the most, you know, important things within Islam, one of the pillars of Islam is Hajj. You go to Hajj, yes, these two are great. Eid al-Futr is great, Ramadan is great. But let's focus on the ayah that Allah, Subhanahu wa Ta'ala, revealed in Al-Ma'idah, verses 67. He says, oh, Prophet, yes, Hajj, prayer, Khoms, Zaka, fasting, whatever, sacrifice to Allah, Subhanahu wa Ta'ala, Eid al-Adha. All of those that he's done within these 23 years, all the battles, the Eids that he had to go through Eid al-Futr and Eid al-Adha, yes, they're great. But if you don't convey the specific message on that day, then everything is worthless.   Narrations do mention that, you know what, before Prophet Muhammad did mention why is it so great, Prophet Muhammad, when he was in Medina, yes, he mentioned to his companions to everyone that every time Ali sat, Prophet Muhammad would stand up, greet him and say, this is my successor. Even if we were to go a bit further back to those who do use that argument, even if you go just a few years back in Al-Mab'at at the number one, the Prophet first began his mission. He asked everyone, everyone had the chance to be the successor of Prophet Muhammad when he says, who is brave enough to be my Wali, to be my successor, to be my close one that wherever I go, he goes. No one stood up. People thought he was crazy. Ali Al-Talib stood up the first time. Prophet Muhammad said, he was 13 or 12. Then Prophet repeated it, he stood up. Repeated it again, stood up. And then at the end, he said, this is my successor. So yes, Prophet Muhammad did mention it way before Ghadir. When he first started Islam, he mentioned that this is going to be, he is going to be my successor, Ali Al-Talib. So why is Ghadir so great? Ghadir is so great because at the time, Muslims were a minority. At that time, Muslims were in the tens. After the migration, they became in the hundreds when they entered Medina. But now they're in thousands. Some narrations say 70, some narrations say 100,000. People were present on that day. So imagine the importance of such a place in such an event where Prophet Muhammad utilizes that time and that place to deliver this sermon. Now it's pretty clear that Eid al-Ghadir is one of the greatest Eids because the axis of Islam really lies on there. And that's where Islam balances out. It balances out on Ghadir. Now if we were just to go and look at some hadiths, try to clear the water up. Prophet Muhammad Sallallahu Alaihi Wasallam, peace and blessings be upon him, and his beloved progeny has said the day of Ghadir is the best Eid for my nation. It's the day on which God the Almighty ordered me to preach the appointment of my brother, Ali Al-Talib, as the flag and principle of my nation. So people shall be guided by him after me. And this is the day in which God has completed the religion and perfected the bounties on my nation and the day on which he was pleased with Islam as their religion. This is actually another verse. Al-Yomu Aqman Tilakum Deenakum Wa Atanakum Ya'amati Wa Adilakum Islam Adina. This is another verse within the Qur'an and a tradition as well. So if we were just to keep that in mind, Allah Subhanahu Wa Ta'ala is telling Prophet Muhammad that today he has perfected the religion of Islam. So let's just comprehend that. On that day, Islam was perfected and Allah chose that religion for the over 100,000 of people who were present on that day. So imagine the greatness of this day and the importance it holds. Now another narration by Imam Sadaq peace and blessings be upon him. He says it is the greatest festival of Allah. Allah has not sent any prophet but that he celebrated this day as Eid and was aware of its sanctity. It is known in the heavens as the day of the famous covenants. You might think that Allah Subhanahu Wa Ta'ala, the Almighty and the majestic has created a day more sacred than that. No by Allah, no by Allah, no by Allah. It's clear that Imam Sadaq and both Prophet Muhammad were telling us that Allah has not created a better day than Ghadeed because Islam was perfected and Prophet Muhammad and Allah Subhanahu Wa Ta'ala chose Islam to be the religion. This hadith can be found in Tadheeb al-Ahkam by Sheikh Atusi Volume 3, verse 143. Just be knowledgeable like me and I'm just kidding. But therefore, as we Muslims, we should never forget the words of Prophet Muhammad.  Yes, have fun, but within Islamic boundaries. But, you know, try to always remember these words and it's important to keep in mind to always tune into hashtag Elinti because we have the freshest episodes with the freshest people. I'll see you for tonight. Wassalamu alaikum warahmatullahi wabarakatuh.


Trini girl natural everyone. Welcome to my channel. I'm just reviewing one product this makeshift styling gel I got it in Target I was kind of being a PJ because most of the mixtape stuff generally have like circles and just stuff I don't put in my hair So when they came up with this supposedly more natural line and it's so I okay What can I try and this kind of popped out as one that wasn't too bad I was a kind of an opportunity to just try mixtape for the first time So I pd and I grab this and I have it in my hair right now So let me tell you about the ingredients and we tell you about the product And then I'm just gonna give you a little review of how it works in my hair. So let's go so Yeah, this is a close-up It's kind of long so it's hard to get everything in there, but mixtape styling gel These are the ingredients like it didn't have any carbomber, but it does have PVP which is along the lines of that kind of synthetic gel But also has some gum in here, which is the kind of gel I like So I was like, well, you know, there's a chance it could be good and then it has like hops and Ivy and lavender and just another cool extra to the bottom So I thought hey, that would be kind of nice to give it a try in terms of this Sand it has like a perfumey scent. It's not super heavy. I could be slightly getting laundry detergent with it, too Like laundry detergent perfume But the scent isn't horrible. So this is the consistency and if you can see that it's a pretty solid Goop as you can see And it is pretty slimy like you see all that So I was kind of excited when I just tried it and like I think if you do it here in small sections It will really clump your curls, but It did dry pretty much similar to how it was wet so didn't like result more dry or anything So I can't say that for it. So soft and nice when it was wet I just the texture of the gel I mean I was like really hoping it wouldn't give me a cast But it is slightly crunchy. So it's not horribly crunchy. It's not like Breaks, but you know, I can feel that cast there. So I know some people like that. But that's definitely a negative for me My hair isn't super dry. So that's good. I Wouldn't even say I guess it's moisturized. I mean, I did use the Sonata Kill slip and slide not proof hair gliding even under it So all it had to do was kind of not dry my hair out and I would say that it didn't dry my hair out So that's good in terms of shine. Definitely not much shine. My hair looks much ashier than normal um There is some definition, but there's also frizz so Some definition some frizz Definitely ashy. I do like the volume. The shape is fine It's not the worst that I ever tried, but it's just not there to my age So I got disconnected and I don't feel like setting up my lighting again. So here I am just doing the end over So basically the last thing I was saying was that I was nervous to try it out because I was getting a lot of um Whispers of people that liking it and I just wasn't seeing that many reviews online Which usually means that a naturally a community that people don't like it YouTubers don't like it. So I was scared to try it, but it really wasn't that bad. So You know, I would say for my target unlike no other good jealous there. Like there's no kimi rules. There's no uncle funky There's no kinky curly. There's no jane carter. I would probably try this. Well, at least I might, you know, like I don't hate it I just don't love it and I don't consider it to be one of my hg. So Yeah, this is uh an interesting video. Let's put it that way But you know, it's all intro business, right? So stay tuned for my next video. I will see you in the next one and bye


That's where we ended last time. We were looking at a minimal area disk diagram, d in x. And remember, that meant that we have a diagram. We have a diagram. And well, we know that our diagram is built from squares. There's a lot of squares. Inside, I'll draw part of a diagram to remind you what's going on over here. And we said we were going to look at dual curves. These were little kind of hyperplanes in the diagram. And we're going to be interested. So there are two dual curves. And we're interested in looking at how the dual curves are going to be wandering around in the diagram. And well, there's a few pathologies that cannot occur when the diagram is minimal. What does it mean for the diagram to be minimal? Well, among all disk diagrams, with a particular boundary path that's that orange path that goes around the disk diagram, we can look at the disk diagram that has as few cells as possible. Usually, it's as few squares as possible, but as few cells as possible is fine. So the minimal diagram or unminimal diagram, they're not necessarily unique, will be one which has as few cells as possible, keeping that orange boundary path fixed. And when the diagram is minimal, the dual curves, and it's a diagram mapping to a non-positively curve cube complex, the dual curves cannot exhibit any of these pathologies. So a dual curve in particular cannot self-cross. What's your question? When you say minimally, do you mean it's an image in the cube path, what does it look like, the D itself? D itself. We're looking at how big D is itself. So when we say a diagram in X, we mean a disk diagram in the plane, together with a combinatorial map from that disk diagram to the two skeleton of X. So let's quickly sketch why this theorem holds. So the idea was hexagon, certain moves, so the method of proof relied on two transformations that allow us to maneuver back and forth between different diagrams with the same boundary path. So this one we like to call a hexagon move, sometimes righto-meister move. And what you're seeing is pushing across a 3-cube. You have three squares. You know that there has to be a 3-cube over there where they map to inside of X. They're going to surround the front half of a 3-cube. And you could just push across that 3-cube and change your diagram. So this is a move that happens inside, maybe could have happened right over here. There's an opportunity to change, to modify that diagram by this hexagon move. And then there's another move, which is a bit more drastic. And it's actually not going to occur inside of a minimal area disk diagram. And it looks like this. We have to be conscious of it. So over here we have two squares in the diagram that are meeting along a pair of edges. And those two squares have to map to the same square. That's why I'm drawing these labels over here. They have to map, they fold on their way to X. Because the cube complex X doesn't have two squares in it that meet along an entire corner. Because what would that mean in the link? Well, it would mean you would have a bygone. If these two squares map to different squares in X, you would have a bygone in the link. But the link is a flag complex. So it doesn't have any bygones in it. So what we can do, if we saw a god forbid, if we saw a pair of squares like this inside of a disk diagram, what we can do is we can remove those two squares and replace the pair of squares by this little arc of length 2. And that reduces the area. So of course that would never happen in a minimal diagram. The diagram couldn't be minimal and have this property. And have a pair of squares like this. In the combinatorial group theory, this sort of thing is called removing a cancelable pair. Those two cells kind of cancel with each other. All right. So now, how does one prove that theorem? Well, the main point is if any pathology exists, if one of those four pathologies existed in the diagram, then there exists a bygone. This is what we would call a bygone. Why is that? It's pretty clear. For instance, if you had this inside the diagram, then you could just choose a dual curve starting somewhere, see where it goes. And either it would self-cross, in which case you'd have one of those pathologies, or you'd find yourself a little bygone somewhere. So an innermost pathology would actually be a bygone. And then what we do is we now transform. So what we're going to do now is we're going to take a bygone. And we're going to show that if there is a bygone, then the diagram wasn't minimal. So let's see how that goes. It's pretty simple. You see, I'm just going to sketch it out. If you had a bygone, we're not saying what's on the inside. It's more squares maybe, or maybe not. If you had a bygone, then with a little bit of work, there's a little bit of extra effort over here, what you could do is you could find a little square corner. I'll give you away in a moment of being sure about that. But you could find a little square corner inside this bygone, just like this. And then once you see that square hanging over there, what you can do is you can push it out of the bygone by using a hexagon move. Get this right. Let me do the following. So you all see that the yellow dual curve, for instance, that was over here, got replaced by that yellow dual curve over there. And what happened is we've now found a new disc diagram with a smaller bygone. And well, you keep doing this until there are no bygones to push out at the bottom or to push out at the top. And at the last stage, you might perhaps keep doing this. At the last stage, you'll have a situation like this. This is also a bygone, isn't it? You've squeezed all of those squares out, though. And now at this point, you can see that if you visualize it the right way, you're looking at the outside of a column of cubes stacked one on top of the other. And you can replace this by this picture over here. And when you do that, let's get this right. 1, 2, 3. OK, I'm not. I don't know if I'm counting correctly. I hope I counted that right. When you do that, you'll reduce the area by 2. Did I get it right? Erase the. Oh, well, that's a separate issue. OK, excuse me, I had a question. What a question. Sorry, I don't get how the first clue occurs. How does what occur? The first clue on the bottom left of the figure. How does this occur? Well, I mean, it might have been that way to begin with. It just turns the image, or? Well, so at the first stage of this process, you find squares that are kind of hanging at the bottom, and you push them out, until there's nothing left on the inside. It might have been that way in the beginning, I don't know. I mean, that's also a bygone, isn't it? But after you've squeezed all the squares out, then you have this. And this is, in some sense, a generalized removal of a cancelable pair, isn't it? Wait, can you just say again what you do in this last step? At this last step. You know, I'm actually very combinatorial minded, right? I gave it to you in one big sweeping. I said, well, imagine this thing. It's a kind of column of cubes stacked one on top of the other. And you're looking at four of the sides of this column of cubes over here. And if you'd like, you can imagine pushing across to the other two sides of the column of cubes. And that's what's happening over here, unless I screwed it up.  OK, let's correct it. But another way of thinking about it, which is maybe better, is you can keep doing hexagon moves until you've pushed one of these squares all the way over to the other square, and you can remove the cancelable pair. OK? So let's do it that way, instead of doing it in, you know, I've done this. And then I have one, two, three more, right? And I've now created a smaller bygone, right? And now you all see, I keep creating smaller and smaller bygones until at the last stage, I can remove a pair of squares, OK? OK, that's better. Thank you for correcting me. Well, before I move on to begin the lecture, because this is really from last time, you can also, if you think through what's happening over here, you can also pick up the furthermore statement, OK? And the furthermore statement says that if it so happens that you can find a diagram that has a configuration like this to dual curves that kind of cross in one spot, and then kind of touch each other, come up right next to each other in another spot, then in the cube complex, there exists a square, S, which could have been positioned right there, OK? This is going to come up again later. But this essentially implies that, right? Because you kind of imagine what's happening over here. And you follow all of these steps, then this square kind of keeps pushing, pushing, pushing until it arrives at the other side. And that shows you that that square exists. Yeah, so I have a question about the first step in this process. What happens if this vertex has four squares meeting there? Which step, right over here? How do you find this? How do you find that? That's a very good question, OK? So you can either work a little bit, OK? Because I hid that. Or you can look at the following proposition. Let D to X be, but you can cook it up. You could think about this on your own and define what it means to be a kind of lowest. You could find the lowest square and you'll find it, OK? But there's another, this proposition, which you can then feed into. This is some type of big induction. Then you'll see that you could find it as well.  Then either D is what we call trivial, right? What's the smallest possible disk diagram that you can have a single point, single vertex? Or D looks like, it's some type of pastry. Maybe someone will know the name of these pastries. Is there anybody here that eats a lot of pastries? So it's surrounded by three dual curves that are meeting at three corners. It's some type of triangle. It's a triangle, triangular. Or D is a triangular pastry, OK? Or the usual case, the number of square corners in D plus twice the number of spurs in D is greater than or equal to 4, OK? So what are those? So a spur in a disk diagram, so maybe I'll give you a picture of that. Here's another example, some more stuff inside. I don't know what. Well, I guess just a single square on its own, right? So a square corner would be like that. And a spur is one of those. So a spur in a disk diagram is when you have an edge sticking out, ending at a vertex of degree one on the boundary. That's called a spur, OK? So here you have a spur right over here. Maybe there was a little tail, I don't know. So the third option is that when you count the square corners, so here I have one, two square corners over there. And I've got a spur over here. Here I have four square corners. This square has contributed four of them, OK? And I leave that for the exercises to understand why this is the case as a consequence of no bygones, OK? I gave a little hint over there of how to do it by a quick induction. And if you don't want to work hard to understand this, then you could set up some induction in your head and say, well, I believe everything. So I believe that proposition. And so show me where the spurs or the square corners are inside of this sub diagram. And there'll be either a spur, which will kind of tell you that you can close it up a little bit somewhere at the end. Or there'll be a square corner, at least one, along the top or bottom. And you'll be happy, OK? I'm more interested, though, in conveying the culture here than elegant systematic proofs. Excuse me? In the further most statement, you said that if there is something in the diagram, then there is a square s you mean in D. The square s is in, thank you, in X at the images of E1, E2. Or in other words, you could have put a square there and mapped it to X. Is it clear now? It's a little wishy-washy? Or it's clear. But how do you know that it exists? You said it's your. If you follow this kind of sketchy proof, then you kind of imagine first pushing things outwards until there's nothing on the inside. Ignore that this was here. We never used it. And then start taking this square and imagine pushing it further and further to the right. At the last stage, after you've done it, after you've pushed it all the way to the right, it will be right alongside these two edges. And that's the square s. What arrives there after pushing, pushing, pushing is the square s that the furthermore is insisting on. But the pushing is building a second or more. Yes, that's correct. But this means that you changed the. It's not this square. In the cube complex, it's a square that's parallel to this square. It's witnessing that there is this column of cubes. And you're just looking at the other at the top end of it instead of the bottom. OK, so you check. OK. OK, so let's start today's lecture now. I want to talk about convexity. So I'm actually going to focus on sub complexes. So y tilde contained in x tilde is convex if gamma is contained in y tilde whenever gamma is a geodesic in x tilde with n points in y tilde. So a remark, we're going to really be very combinatorial over here. When y tilde is a sub complex, not just a subspace, it turns out that y tilde is convex, provided that geodesics gamma joining vertices of y tilde must lie in y tilde. And also, one other thing, a cube is contained in y tilde whenever its one skeleton is contained in y tilde for each cube of dimension greater than 0. OK, so you have to think about this a little bit. But what I'm saying is that maybe I'll draw an example of a convex complex so that you'll have something to have in mind. It's the one skeleton of y tilde. Thank you, dear. It's just a very simple example. So that is y tilde over here. And it lives inside of x tilde. And the statement, you should have the remark in mind, I'm really focusing on the main point are geodesics joining vertices in the one skeleton. So geodesics in the one skeleton joining vertices. Excuse me. So convexity is saying you can't have a geodesic, like this orange one, which starts and ends inside of y tilde and leaves y tilde. OK, yes? In the first part of the remark for convexity, if I take a three cube, I don't think it's one thing. I remove it. It only takes three of the two cubes in its boundary. It's convex how long it has to be the entire cube. Let's see. So that's correct. Did I make a mistake? Let me understand what you're asking about. Three cube, that is three of the faces. Now, not the entire one skeleton is inside that sub-context. OK, so here's my three cube. And what are you asking about? Essentially, if I take only three of the faces, three connected faces of them, the convex hull of that is going to be the entire cube. That's correct. But the one skeleton is not going to be the concrete. Yeah, you're right. Let me fix the statement. But it still has to be the entire cube because you need every geodesic joining two corners. So he's complaining that I made a mistake. OK, so let's correct it. It should say whenever a corner of a cube lies inside. So how am I going to crack that? It's a little painfully.  Thank you for saving me over there. So what's our mean in the picture? This one, yes? It is convex. And the orange path is not a geodesic. So this is convex. And this is not a geodesic. It's a picture of failure. Now, probably the main point over here is the following definition. A map from y to x between non-positively curved cube complexes is a local isometry if it is a combinatorial map. So it sends cubes to cubes if it is combinatorial. And for each 0 cube, little y, the induced map from the link of y to let's name little x, where y maps to, the induced map from the link of y to the link of x is an embedding as a full sub-complex. So the, well, when you imagine, here's a simple example.  So this is R2 mapping to R3 over here, very poorly drawn. And what's happening at the level of the links? Here's the 0 cube that I'm interested in. That's little y. And here is its image, little x. And at the level of links, the link of little x looks like this red octahedron. So this is the link of little y contained in the link of little x. And to be a full sub, so there's two things going on over here, there's the embedding. So that's really saying that this is a locally injective map. So the vertices of this link, the vertices of the link of this point are mapping to vertices of the link of that point. But it wouldn't necessarily be an embedding. So we're requiring that it's an embedding. And we're also requiring that it is full. So whenever a simplex, whenever a set of vertices in the image of link y span a simplex of the link of x, that simplex has to lie inside of the link of y. That's what a full sub-complex means.  If you have a subset of the vertices, you throw an edge in whenever its vertices are in the set. That's going to be a full subgraph. A full sub-complex of a flag complex, or of a simplex complex, actually, a full sub-complex will have the property. What it means is that whenever the vertices of a simplex are in the sub-complex, the simplex is in the sub-complex. So there's a similar variant version of this, which I find quite convenient. Maybe I'll call this an exercise. Equivalently, the map is a local isometry if it is a combinatorial immersion. So now I'm asking that it be locally injective. This just means locally injective. And it has the following property. There are no missing corners of squares. What does that mean? What it means is that whenever two edges, E1 and E2, this is inside of y, whenever you have two edges, E1 and E2 and y, at some vertex of y, maybe I'll call that little y, actually, that's what we were calling y before. Whenever the image of this pair of edges, let's call them now E1 bar and E2 bar, this is inside of x and v bar is the image of v, whenever these two edges at a vertex bound a square, whenever they bound a square, let's call that square s bar inside of x, they actually had to bound a square inside of y. So here's the square, maybe I'll draw it in yellow, here's the square that actually has to exist. So let's stare at this for a moment. What this says is that if two edges at the vertex v in y map to two edges at a vertex in x, which form the corner of a square, which I'll call s bar, then those two edges actually already form the corner of a square s, which of course maps to s bar. So there's no missing corner of squares. If there could have been a square there, there was a square there. Now, this is somehow much more specific. It's much more specific than this definition over here. It just seems to be focusing on one small tiny little point. But it turns out when you think it through, they're equivalent and very heavily depending upon that these were not impossibly curved cube complexes. So a quick example, any immersion, really combinatorial immersion of graphs is a local isometry. And in some sense, we're always generalizing things that work for graphs. So when you have a local isometry of graphs, people usually call them just an immersion. People usually just say an immersion of graphs. It's a local isometry. Well, there are no squares around, so that's automatically the case. I will describe, so in the exercises, there'll be more discussions of local isometries. Let's state the theorem. Let phi from y to x be a local isometry with y and x connected, of course, all cube complexes over here are connected, non-positively curved cube complexes. Then number one, the induced map between their universal covers is an injection. Number two, consequently, the map phi induces an injection from pi 1 of y to pi 1 of x. And actually, a bit stronger than number one. In fact, the lift of the universal cover of x to the universal cover of y to the universal cover of x is actually not just injective, but its image is a convex sub-complex. So let's sketch why these are true, and it'll give us a chance to see our disc diagrams in action. So I think let's start by proving injectivity. Suppose that A is not equal to B are two distinct points of y tilde, and suppose that they are mapped to same point of x tilde. Since these are complexes, it's enough to show that this map is injective, it's enough to show that it's injective on zero cells. So we will assume that these are vertices. And now we will choose a path P to y tilde joining A to B. This is a combinatorial path. So it's a path in the one skeleton. And actually, we're going to choose that path such that P is equal to the boundary path of a disc diagram D, where D to x is a disc diagram. And moreover, let's do it so that D is minimal among all possible such choices. Well, yes. But right. But it's closed in the disc diagram, which is mapping to x. So here's A, here's B, here's P. Right now, they're different from each other. A and B are different from each other. In x, of course, A and B are going to map to the same point because they're the same point in x tilde. And so if there's a path from P to x tilde, which if there's a path from P to x tilde, which is closed, so the path from P to inside of y tilde, of course, A and B are different from each other. But inside of x tilde, they're actually the same. The images are the same. So I should always say A bar and B bar. That would have clarified things, sorry about that. And so since this is a closed path in x tilde, which is simply connected, there's going to be a disc diagram. And well, we like minimal disc diagrams, so let's choose the smallest possible disc diagram. And let's even, we don't really care which choice of P we're going to use from A to B. Whoops, from A to B. We'll choose whichever one would give us the smallest disc diagram. You mean D to x tilde or D to x? Well, it doesn't matter, of course. If you have a disc diagram in x tilde, it projects to a disc diagram in x. If you had a disc diagram in x, it would lift to a disc diagram in x tilde. Since D is simply connected. x tilde. That's fine, thank you. And now what do we know? We know that disc diagrams have spurs or corners or they're trivial, right? So if D is trivial, then A equals B because P is the trivial path, right? So that couldn't possibly happen. Otherwise, by the proposition D has a spur, well, it has at least two spurs or square corners, right? The proposition up over there, yeah, the triangle has three square corners, right? So there's one of them. Can't reach the others. OK? But what would that mean? Let's just look at the case of a square. Let's say that D had a square right over here. Since there are at least two, there's going to be one which is going to be away from this point right over here, where A and B kind of get identified. So there's going to be at least one away from this point. So that means that there's going to be a square kind of alongside y, alongside y tilde if you'd like. So that square is going to be kind of like this. But y tilde doesn't have any missing corners of squares. So this yellow square actually must live inside of y tilde, right? But that means that instead of using this diagram, we could choose a different this diagram. We could use this diagram over here, D prime, right? And of course, we'd be choosing a different, we'd be choosing P prime instead. And this is smaller, right? In the case of a spur, you'll do something similar. All right. So I guess we already have injectivity of the map between universal covers, which implies pylone injectivity as well. How about convexity? Yeah? How do you know that there is a minimal disk diagram that's in the image of y, sort of? How do I know that there exists a minimal disk diagram? There's boundary facts in the image of y. So we chose the path p inside of y, or inside of y tilde. That path p is closed inside of x tilde. Therefore, it's no homotopic because x tilde is simply connected. Simply connected diagrams have disk diagrams. We chose the smallest one. So let's look at item three now. So let's prove convexity. So that was three. Now we're going to do something very similar. Let gamma be contained in x tilde via geodesic from A to B. So y tilde already embeds. So I could just call these A and B now. And we're going to choose a geodesic from A to B. And let D be a disk diagram between gamma and a path, a combinatorial path sigma, such that D is minimal among all choices fixing gamma. So the picture is this. This is what I mean by a disk diagram between gamma and sigma, a disk diagram whose boundary path is gamma sigma inverse. And this is all the territory of y tilde over here. I'll draw it in yellow. That's y tilde. OK, so now it's quite similar. It's going to be a very similar style of reasoning. A little more elaborate though. But the basic idea is to choose this minimal area diagram and then use things that we know about dual curves now. So firstly, no dual curve starts and ends on gamma. So let me draw the picture of what can't happen. So here's a dual curve that kind of travels from gamma to itself. So this cannot happen. Why not? Well, if you imagine something like that, what you would do is you would say, well, consider each dual curve that travels through, that crosses the ladder carrying this dual curve. And where would it end? So I was kind of started over here. Where would it end? It has to end on this little part of gamma in between. Because could it come back and cross the orange? No, because a minimal diagram can't have any bygones. So this orange curve has to end on gamma. It can't come back there because then we would have a bygone and then d wouldn't be minimal. So all of the dual curves, I'm just going to draw some of them. All of them end on gamma. But what does that mean? That means that this part of gamma in between these two points is at least as long as this part right over here of the rail of the ladder of the dual curve. So what it means is that this path over here is shorter than gamma. Since then, gamma prime is less than gamma. It's two edges less. It's too less. Because it missed this edge, which is dual to the orange curve, and it missed this edge as well. And this part is no longer than this part. And the second step of the proof is the following. Well, if d contains a square, here's gamma, here's sigma. This is d. If it contains a square, what we want to show is that d doesn't contain any squares. So gamma actually lies, gamma will be forced to lie inside of y tilde. If d contained a square, well, we might as well choose the first edge which doesn't lie. We might as well choose the first edge of gamma which isn't equal to sigma. We might as well choose the first edge where gamma leaves y tilde. We might as well have assumed that gamma actually left y tilde immediately and arrives immediately at y tilde. So where does this square end up? It can't self-cross. We already said it can't come back to gamma. So it has to, if you look at the dual curve that sort of starts at this square, it has to end on sigma. I already said it can't come back. So it has to cut all the way through to sigma. OK? Well, now apply the proposition to this little sub diagram over here. We've got to have at least one more corner someplace. There's a, sure, maybe there's a corner over here and there's a corner over there, but there's got to be a corner or spur someplace else. Perhaps over here, because there's going to be at least three corners in this diagram. And then the same exact thing that happened before. This square would actually lie inside of y tilde, because y tilde doesn't have any missing squares. So the upshot is that d is trivial. Gamma and sigma are the same, and we're done. I have to stop. So that means you guys get to go and have your 15 minutes of coffee. Am I wrong? Yeah. Thank you. Thank you.