...Algebraic and combinatorial techniques of knot theory based on arc colouring By Ayodele Arubi Advisor: Dr. Alexei Vernitski A project submitted in partial fulfilment of the requirements for the Degree of Bachelor of Science with Honours in Mathematics University of Essex Colchester, Essex April 2015 Contents Abstract 4 Dedication 4 Acknowledgments 4 Introduction 5 The history of Knot theory 5 Brief history of knot theory 5 Development of the Knot Theory in Physics 7 Counting knots 9 The modern knot theory 12 Defining a Knot 14 Wild Knot 15 Definition of a Knot 16 Differentiating Knots 18 Orientation 18 Knot arithmetic 19 Modular arithmetic 20 Equivalence relation 21 Additivity Property 22 Multplicitivy Property 23 Knot Invariants 24 Reidmeister moves 24 Fox colouring 26 Dehn Colouring 28 Alexander-Briggs colourings 30 Implications of knot colourings 31 Other Knot invariants 31 The linking number 31 Quandles and Racks 35 Brief history about the Rack and Quandle invariant 35 Kei 36 Quandle 37 Racks 38 Definition 38 Examples of racks 39 Important definition of Racks 42 The free product. 42 The Cartesian product. 42 The disjoint union 42 Orbits and stabilizers of racks. 43 The operator group 43 The associated group 43 Bibliography 46 Abstract This project narratively examines the history of the knot theory, its invariants and the theory of arc-colouring. This project will begin...
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...Balkan frame -2 horizontal bars -diagonal bar -4 vertical bars -3pulleys -Clamps -Overhead trapeze -Cross bar -Firm mattress -Fracture board -Shock blocks / lock b.BST equipments -Thomas splint -Pearson attachment -Rest splint -Cord/Sash -Foot rest -Safety pins/ paper clips -Thigh rope(shortest) -Suspension rope(longest) -Traction rope(longer) -Traction weight-Suspension weight 4. Traction Set-up a. Thomas splint and pearson splint 1. Attach the rest splint to the Thomas splint with Pearson attachment 2. Upper part is the Thomas splint which will support the thigh and lower part is the Pearson attachmentthat will support the leg. 3. Tie the short rope to the medial upright of the Thomas splint with slip-knot to ensure privacy to thepatient b. Application of slings to the Thomas splint and Pearson attachment. 1. Start from the...
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...Gidion’s Knot Gidion’s Knot was one of the most intense live performances I have ever seen, probably because it was also my first. Right off the bat I noticed that a few audience members were sitting in the school desks that I originally thought were part of the set, and they were, but audience members were allowed to sit in them. Though I did not, I thought it was a cool way for people to truly feel apart of the play. Between that and student work posted around the walls, you really felt like you were in a kids classroom. The play started with Heather, the teacher, at her desk, grading papers but seeming preoccupied and upset. Upon Corryn’s entrance, Heathers dialogue seemed very static and plain, as though reading from the script. She seemed to be relying on heavy sighs and breathing for emotion. I felt immediately this might be a one-sided performance but her speech rhythm began to loosen and flow more naturally the more she spoke. The more Heather talked the more you think she knows about Gidions suspension. She seemed to have knowledge of the issue, but plays the defensive as if she’s uncomfortable saying, or just trying to keep the conference professional and appropriate. The actor did a good job keeping that demeanor, her posture straight, face blank when possible, body language cordial but reserved and constrained. After understanding...
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...This publication is available at Army Knowledge Online (www.us.army.mil) and General Dennis J. Reimer Training and Doctrine Digital Library at (http://www.train.army.mil). *TC 21-24 Training Circular No. TC 21-24 Headquarters Department of the Army Washington, DC, 9 January 2008 Rappelling Contents Page PREFACE..............................................................................................................vi Chapter 1 TOWER RAPPELLING ...................................................................................... 1-1 Section I - Personnel ........................................................................................ 1-1 Rappel Master .................................................................................................... 1-1 Rappel Safety Officer ......................................................................................... 1-2 Rappel Lane NCO .............................................................................................. 1-3 Rappeller ............................................................................................................ 1-3 Belayer................................................................................................................ 1-4 Belay Safety ....................................................................................................... 1-4 Section II - Preoperations Briefings and Safety Procedures ....................... 1-4 Safety......................
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...Virtual Spatial Graph Theory 1. Introduction The mathematical theory of knots studies the many ways a single loop can be tangled up in space. Since many biological molecules, such as DNA, often form loops, knot theory has been applied to biological systems with good effect. However, many biological molecules form far more complicated shapes than simple loops; proteins, for example, often contain extensive crosslinking between cystine residues, and hence from the mathematical viewpoint are far more complicated structures–spatial graphs. The study of graphs embedded in space is known as spatial graph theory, and researchers such as Flapan have obtained good results by applying it to chemical problems. However, in biological systems, proteins are often associated with membranes, meaning that some portions of the molecule are prevented from interacting with others. In the case of a simple loop, the virtual knot theory of Kauffman provides a mathematical framework for studying such systems, as it allows some crossings of strands to be labeled “virtual,” i.e. non-interacting. We hope that a merging of these two theories, called virtual spatial graph theory, will prove equally useful in the biological sciences. Knot theory studies embeddings of circles up to isotopy. There are many ways to extend the ideas of knot theory; two natural choices are the study of spatial graphs and the theory of virtual knots. The theory of spatial graphs generalizes the objects we embed, by studying...
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...Analysis of Untying the Knot Analysis of Untying the Knot Introduction to Sociology | Alan Smith Introduction to Sociology | Alan Smith Tarri Boyd Tarri Boyd 2015 2015 Tarri C. Boyd March 5, 2015 Introduction to Sociology Professor Alan Smith Analysis of Untying the Knot The article “Untying the Knot” by Melanie Thernstrom is about the story of a once happily married couple Max and Kate going through divorce. Kate and Max was the ideal couple in the beginning having cool interests, cuteness, similarities and differences that attracted them to each other in a unique way. They are equally attractive, in differing genres (Thernstrom, 2003 pg. 38). Kate felt as though a new world was opening through her very eyes upon meeting Max, they both enjoyed learning new hobbies, experiences and tastes. Kate and Max’s marriage is an example of exogamy, which is marriage outside the group and its culture (Conley, 2013 pg. 376). Usually in societies like the west, marriages outside the social group is legal and perhaps socially acceptable as well now than it was in the early twentieth century. Since the United States is so culturally diverse it’s very common now for people marry other races of people and have interest in their ethnic ways. In the case Max and Kate, Max was Jewish and after seeing each other Kate converted to Judaism. They both got involved in in a local temple, and five years after they met, when they were both 36, Kate converted to Judaism (Thernstrom...
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...Teaching Analysis HOW TO TIE A WINDSOR KNOT Core Concept I chose to write a paper on how to tie a tie (The Windsor Knot). I chose this video because I wanted to compare it with the way I tie a tie. This particular video used the double Windsor knot. When I started watching the video I didn’t realize it was for the double Windsor knot. There are several ways to tie a tie and I find myself looking at other people’s ties to figure out if they did it the same way that I do or what method they use. Some people’s ties look a little crooked or sloppy and I always wonder what approach they took when they tied their tie. I always wonder if they may have left a step out. Some people’s ties are too short so I figure that they don’t know how to adjust the tie. I like my tie to be straight and the right length. I am a tall person so I have to make sure I tie mine a little longer. When I meet someone dressed up and wearing a tie, the tie is one of the first things I notice. So I assume that other people are noticing the same thing and this always makes me cautious of how my tie looks before leaving my house. Organization I feel like this video (How to Tie A Tie For Begginers) was well organized. He demonstrated the tying process very well. He started off with a tie around his neck and explained step by step what to do. Even to the point of what end of the tie should be in the right hand and which end should be in the left hand. If this was my first time at tying a tie, I believe I...
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...Caniel Snowball 6/17/15 The Shipping News is a story about letting go of the past, the pursuit of happiness and a bunch of knots. Annie Proulx’s clever use of symbolism and interesting array of characters cohesively make an impactful tale and string the storyline together. A theme of having a troubled past becomes clear as it tactfully explained throughout the story to project potential for the growth. Quoyle is the story’s main protagonist who has a mid-life crisis at 36 when his cheating wife dies in a car accident and his parents commit suicide. Proulx describes Quoyle’s past as one with many shortcomings paired with his awkward looking chin and size that encapsulate him in a shell of low self-confidence. His newly found Aunt Agnis appears in his life after his parent’s deaths to bring him to Newfoundland for a new start on life. Quoyle find love and courage in Newfoundland and comes to point where his wife’s death doesn’t haunt him. The aunt, a widowed hard alcohol drinking closet lesbian, throughout the book attempts to stay busy with work and making plans. Upon introduction, Agnis appears to be a flat character but progresses into a more complex character when Quoyle’s father’s incestuous indiscretions with his sister are revealed. The reader can retrieve a sense of how she fits the operating method of someone after a sexual assault. Agnis’s refusal to show weakness or vulnerability and constant need to show she is in control adds depth to her character. When Cousin...
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...OUROBOROS DJD5-assesment 2 By Raegan Bregmen Student id: s2014000042 Introduction This essay is about Ouroboros .i will be researching multiple various cultures that I’ll be searching into trying to find its relevance for today’s age and the impact it has on todays stone carving culture. What is Ouroboros? The Ouroboros is a greek work for tail devourer and is one of the oldest mystical symbols in the world. This symbolises the cyclic nature of the universe; creation out of destruction, life out of death. It unites opposite; the conscious mind. What does Ouroborus Mean? -Self reflexivity – the represents something constantly recreating itself -Cyclicality – it reoccurs or revolves in cycles -Primordial unity – this represents something with great force and qualities that can’t be extinguished -Death-rebirth – represents him eating himself, impregnation himself and giving birth to himself -The eternal return – this represent them returning to life as soon as they pass on -Re-creating itself – represents death then rebuilding yourself -Harmony of opposites - to make everything exist in balance and harmony. The above are two different versions of the term ouroboros On this page I have tried to include some information about the history and my understanding of the meaning of the ouroboros The circular natured form represents death and life of each life born. The head of the serpent symbolises life and the rear end of the body represents the end of life. In some...
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...Blood Knot Essay. In Blood Knots, Mallory Burton develops the idea that an individual’s personal problems must be dealt with otherwise it will affect them in negative way. Unless personal problems are kept under control, a person's chances of achieving career and personal success diminish. She was going through grief when her father died. Mallory and her dad were the only people who loved to fish. She wanted to go on fishing on the day of funeral as her mother was handling everything fine. Later on she went for fishing with the gear that her dad had as it was cool and stunning looking gear. Later on in the end she didn’t even come for the funeral of her dad. She wanted to go on fishing on the day of funeral as her mother was handling everything fine. At first her mother denied for letting her go for fishing alone and said that i told you maybe to go for a walk. At the same her neighbour was also there and she also saw the beautiful gear that was owned by Mallory’s Father. The neighbour also insisted her to let her go for fishing or else she will also be sad if she sits in a house for the whole day. For some time her mother didn't let her go and said that instead of going fishing prepare for the funeral. When her neighbour...
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...cravat. Neck ties are generally unsized, but may be available in a longer size. Men and boys wear neckties as part of regular office attire or formal wear. Neckties are traditionally worn with the top shirt button fastened, and the tie knot resting comfortably between the collar points. However, it has become common in recent times for neckties to be worn as a casual item, tied loosely around the neck, nearly always with one or several buttons unfastened. and Now, I want to tell you about one method of tying a necktie. named "Four-in-hand knot" The four-in-hand knot is a method of tying a necktie. Also known as a simple knot or schoolboy knot, the four-in-hand is believed to be the most popular method of tying ties due to its simplicity and style. The four-in-hand knot is tied by placing the tie around the neck and crossing the broad end of the tie in front of the narrow end. The broad end is folded behind the narrow end and brought forward on the opposite side, passed across the front horizontally, folded behind the narrow end again, brought over the top of the knot from behind, tucked behind the horizontal pass, and the knot pulled snug. The knot is slid up the narrow end of the tie until snug against the collar. The four-in-hand knot is the prescribed manner of wear for officers of the United States Navy in full and service dress blue...
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...This article has been cited by other articles in PMC. Go to: ------------------------------------------------- Abstract The ileosigmoid knot is an uncommon but life-threatening cause of closed loop intestinal obstruction. Its treatment is different from a simple volvulus in that it has to be operated upon immediately. Preoperative CT scan diagnosis and prompt treatment can lead to a good outcome. Findings of simultaneous ileal and sigmoid ischemia with non-ischemic colon interposed in between should, in an appropriate clinical setting, indicate this condition. The presence of the whirl sign, medially deviated distal descending colon and cecum, and mesenteric vascular structures from the superior mesenteric vessels that converge toward the sigmoid colon on CT scan help clinch the diagnosis. Keywords: Compound volvulus, ileosigmoid knot, intestinal obstruction Go to: ------------------------------------------------- Introduction Ileosigmoid knot (also known as compound volvulus) is an unusual and rare cause of intestinal obstruction.[1,2] In this condition, loops of ileum wrap around the base of a redundant sigmoid colon to form a “knot”. The condition progresses rapidly to gangrene, so early diagnosis and operative treatment are vital. CT scan plays a major role in the evaluation of any acute abdomen.[3] We describe the CT scan features of this rare condition, the awareness of which can help achieve a preoperative diagnosis. Go to: ------------------------------------------------- ...
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...enabling woody plants to grow large or to stand up by themselves. It also mediates the transfer of water and nutrients to the leaves and other growing tissues. Wood may also refer to other plant materials with comparable properties, and to material engineered from wood, or wood chips or fiber. The Earth contains about one trillion tonnes of wood, which grows at a rate of 10 billion tonnes per year. As an abundant, carbon-neutral renewable resource, woody materials have been of intense interest as a source of renewable energy. In 1991, approximately 3.5 billion cubic meters of wood were harvested. Dominant uses were for furniture and building construction.[2] Contents [hide] 1 History 2 Physical properties 2.1 Growth rings 2.2 Knots 2.3 Heartwood and sapwood 2.4 Color 2.5 Water content 2.6 Structure 2.6.1 Earlywood and latewood in softwood 2.6.2 Earlywood and latewood in ring-porous woods 2.6.3 Earlywood and latewood in diffuse-porous woods 2.7...
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...each year through the #1 wedding website TheKnot.com. Other products from The Knot include The Knot national and local magazines, The Knot books (published by Random House and Chronicle) and television programming bearing The Knot name (aired on the Style and Comcast Networks). The Company also owns WeddingChannel.com, the most visited wedding gift registry website. The Company's brand, The Nest focuses on the newlywed-to-pregnancy lifestage, with the popular lifestyle website TheNest.com, a home decor book series with Clarkson Potter, The Nest magazine and baby offshoot The Bump and LilaGuide. Also under The Knot, Inc. umbrella are WeddingTracker.com; GiftRegistryLocator.com; party planning site PartySpot.com; teen-oriented PromSpot.com; and local baby services and community site LilaGuide.com. The Knot, Inc. is based in New York and has several other offices across the country. Word of mouth Today Show Oprah Building relationships Brand ownership Expert on specif subject XO Group Inc., formerly The Knot Inc., was founded in 1996 by four good friends, two of whom had barely survived planning their own wedding due to the lack of local information and real-world resources available. With firsthand experience, a unique vision, and funding from America Online, they set out to reinvent the way couples plan their weddings -- and life’s most important celebrations. XO Group's first site, The Knot, quickly became the most popular online wedding resource and a highly successful...
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...Executive Summary XO Group Inc. is an internet based lifestage media company. XO Group has four flagship brands and each brand is specific to a particular market. The Knot and Wedding Channel are focused on the wedding industry. The Nest is dedicated to the home environment. The Bump is focused on family planning. XO Group provides services to couples who seek advice and information on planning lives together; from the wedding to their first baby and everything in between. XO Group’s professional staff ensures goals are being met and customers are satisfied. Throughout the company’s life, XO Group continually pursues new opportunities to stay ahead of their competitors and hold their place at the top of the lifestage media industry. Table of Contents Page No. | Executive Summary………………………………………………………………. | 3 | Table of Contents…………………………………………………………………. | 4 | List of Tables……………………………………………………………………... | 6 | List of Figures…………………………………………………………………….. | 7 | Section 1: History of Company………………………………………………….... | 8 | The Beginning…………………………………………………………………. Getting Bigger…………………………………………………………………. The Nest………………………………………………………………………... Merging Two Companies……………………………………………………… Stepping Into The Baby Market……………………………………………….. XO Group Inc.…………………………………………………………………. | 8...
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