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Encyclopedia of knot theory.

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Colin Adams, Erica Flapan, Allison Henrich, Louis H. Kauffman, Lewis D. Ludwig, and Sam Nelson

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

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Jessica Purcell is a professor of mathematics at Monash University in Australia.  Her research is in the area of hyperbolic geometry and knot theory.

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Knot Theory (190), Winter 2019

My office hours are  Wednesday and Friday 11-12 (right after class) in AP&M 7210 .

Homeworks will normally be due on the Monday following the section at 4pm . There's a dropbox in the AP&M basement. Midterm will be in class on Friday Feb 8 . F inal is on Friday March 22nd at 8am in ****.

Grading will be 30% homework, 20% midterm, 50% final.

Knot theory

To prove this we need to apply some algebraic topology (e.g. the fundamental group of the complement of the knot in space), geometric topology (e.g. looking at surfaces associated to the knot - we will spend some time on the topological classification of surfaces in its own right), combinatorial topology (e.g. counting 3-colourings of a diagram, or the famous Jones polynomial) or other cleverness. The study of knots is both a testbed in which to apply the abstract theory of topology, and a source of new problems and methods. Plus of course it's fun! (Challenge: make a trefoil as above, and then make one where all the overcrossings go under and vice versa. Are they the same?) 

Current version of knotes, section 1

Current version of knotes, section 2

Current version of knotes, section 3

Current version of knotes, section 4

Current version of knotes, section 5

Current version of knotes, section 6 (The part about topological spaces and surfaces will not really be needed in what we do next, but it may be helpful.)

Current version of knotes, section 7

Brief notes on point-set topology

C. Adams, The Knot Book (1994, W. H. Freeman) This book is a survey of knot theory. It isn't a typical textbook - it is not very detailed mathematically, since it's actually aimed at clever high-school students! But it does attempt to give the flavour of some really quite advanced topics, including current research and open problems!

M. Armstrong, Basic Topology (1983, Springer-Verlag). This is a nice undergrad-level book which teaches point-set topology and the foundations of algebraic topology. It's not directly relevant to the course, but if you are interested in point-set topology, this is probably the best place to look.

N. Gilbert and T.Porter, Knots and Surfaces (1994, OUP) An undergrad-level book, which as I remember contains some basic point-set topology too. Its focus is extremely algebraic, however - it goes into group-theoretic aspects in a lot of detail.

W. B. R. Lickorish, An Introduction to Knot Theory (1997, Springer GTM) The "new testament" of knot theory, a graduate-level textbook dealing especially with post-Jones-polynomial knot theory. It's by my PhD advisor - you might enjoy his dry wit!

There are lots of knot theory resources on the web these days - these are the main ones that spring to mind. You'll learn a lot just by surfing these sites, and they also provide tables of calculations of pretty much every knot invariant you can think of, some software for drawing and calculating with knots, and beautiful images.

The Knot Atlas (wiki) The KnotPlot Site

Table of Knot Invariants

Pre-requisites, and comments about 190 and 191 courses

Traditionally, we teach the more geometric, visual side of the subject after teaching all the basic tools. This is not unreasonable, but it does take a long time to do properly, and is quite hard to motivate because it turns history on its head. After all, people have been using and thinking about knots for thousands of years, but the definition of a topological space is only a hundred years old.

Fortunately it isn't necessary to work this way round. With a little care we can do quite a lot of knot theory without needing to talk about the foundational aspects of topology.

The follow on Math 191 course will be more of a "traditional" algebraic topology course.

Homework problems

One of the common problems faced by students in topology is deciding how much detail to write in proofs; the subject spans a great range, from the most pedantic and precise point-set arguments, to visual arguments which can seem like ``hand-waving''. I hope that the course will help you in general to appreciate and produce ``good mathematics'' at whatever level is appropriate.

You may find the homework questions a bit strange to start with... DON'T PANIC! Here are a few comments.

1. They are not like calculus problems, where you just manipulate formulae and write "equals, equals, equals" down the left-hand side of the page! The idea is generally to prove things.

2. A proof will often amount to just presenting a logical argument or explanation in English . Don't be afraid to write plenty of words - just make sure that they are clear meaningful words, and not waffle! You may feel that a verbal argument isn't really mathematical, but this is not true: maths is about the precise communication of precise ideas, and they don't always need to contain formulae and funny symbols.

3. Please try to write coherently! One helpful tip is to keep a particular reader in mind: imagine that you are trying to convince a fellow member of the class that something is true, and that they will not necessarily `know what you mean' if you write unclear, vague and confusing things, and will argue with you if there are gaps in what you say.

4. In a high-level subject like topology, you often have to use a bit of judgment to decide how much detail to put into your argument; this comes mostly with experience. If, for example, you find you need to appeal to some `obvious' fact,first ask yourself whether it really is obvious! Are you confident that you could prove it if challenged? If so, it's probably OK to just say that you are using it, and not bother writing its proof. But if you have no idea at all how to prove it, then it may well in fact not be true, and you should be wary! (If you can't see any way of doing without the `fact', you can start off by saying "Assuming it is true that..."; that way,your proof will still be true, even if the hypothesis you need isn't!)

Homework 1, due Tuesday 22nd Jan

Read through section 1 of the "knotes" and do exercises 1.2.6, 1.2.9, 1.2.10, 1.4.3, 1.6.2, 1.6.3, 1.6.7,  1.7.1, 1.7.2. (This first homework consists almost entirely of "problems to make you think", rather than "this will be on the exam"-type problems, so don't panic, treat it as an exercise in playing around as an important part of mathematical research!)

Homework 2, due Monday Jan 29

Homework 3, due Monday Feb 4

Homework 4, due Monday Feb 11

Some sample solutions are here . I tried to write these out as completely as I could. The first one is necessarily very long, since the point is  never to just assert  things like "these are the only possible diagrams which..." without proof. For small numbers of crossings like this, it's actually not too hard to just write down a list of all possibilities and convince yourself intuitively that it's complete. But if we tried more crossings,  it would become harder and harder not to miss a few, or to convince yourself that you'd got all of them. This overall style of proof would now be essential - but in writing it down, we would typically now omit a lot of the smallest details so as not to clutter up the "higher-level" overall argument with proofs of things that everyone can just "see" are true. In the second one, I wrote down some details of modular arithmetic which I normally would have omitted (and which you can too if you're happy with it). The final one is pretty much the sort of argument I'd expect you to be able to write.  With experience, you should become better at deciding  how much detail to include,  and how much to leave out.

It will cover everything we've done up to and including the HW above, but nothing further. Since youwon't have met with Katie before the MT, I will keep any questions about 3-colourings quite basic; there will be nothing involvingmatrices. Please bring a blue book . For practice, look at the more concrete exercises in the knotes, especially those marked  something like "2007M" (which are from old midterms), and at the multiple-choice questions at the endof chapter 3. And don't worry, you will not have to answer philosophical questions like1.7.1, or draw long sequences of Reidemeister moves!

Knot Theory

The website for learning more about knots.

One thing that makes Knot Theory so interesting for mathematicians today is the fact that it’s such a “new” topic – Knot Theory is a relatively young field with many opportunities for discovery and exploration by mathematicians young and old. From this website you can learn:

A Brief History of Knot Theory

From the mathematical surge of interest in knots a little over a century ago to the recent and exciting application of Knot Theory to DNA and synthetic chemistry, you can get an overview of why knots are such a fascination for scientists and mathematicians alike.

Introduction to Knots

Learn what’s different about a “mathematical” knot and how we work with, transform, and classify knots. Also, find out what is the central problem of Knot Theory and what properties knots have. This page also has a brief introduction to links, another important topic in the theory of knots.

Advanced Knot Theory Topics

Once you understand the concepts in the introduction to knots, this page expands your knowledge with connected sums, composite and prime knots, stick knots, wild knots, and even has a section on coloring knots and links and why coloring is such an important topic to mathematicians.

DNA and Knot Theory Today

The recent interest in knots has been fueled by discoveries that involve knotting in the DNA strand, the genetic code that resides in all living things. Geneticists have discovered enzymes that actually “unknot” the DNA strand so that it can replicate. What interest scientists the most are the changes that occur in the structure of the DNA strand as a result of these knottings and unknottings.

Activities to Get Your Hands on Knot Theory

From an interactive game involving simplifying knots visually to 3D virtual reality knots to classroom-tested lesson plans, this page offers you a wide variety of ways to visualize and manipulate knots to extend your understanding of the topic.

“Knotty” Fun for All

Fun facts, game-like activities, jokes, links to other great knot sites, and a small picture gallery of Mr. Payne’s knot photographs can be explored on this enjoyable and informative entertainment page while you are learning about Knot Theory in other areas of the site.

If you have ideas, questions, or comments about the Knot Theory Online site and would like to share them, please feel free to e-mail  Mr. Payne  or Dr. Nardo . We are especially interested in hearing from teachers who have used knots in the classroom and students who have been motivated to study knots either in the classroom or on their own.

This site was developed by Bryson R. Payne, M.Ed. student at North Georgia College and State University  in fulfillment of Math 7090 with the assistance of supervising professor Dr. John Nardo. The opinions  expressed in these pages are those of the author, not the university. Please direct all inquiries to Mr. Payne  at [email protected] or visit his personal web site at www.freelearning.com . Thank you!

Teaching and Learning of Knot Theory in School Mathematics

• © 2012
• Akio Kawauchi 0 ,
• Tomoko Yanagimoto 1

Osaka City University, Osaka City, Japan

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Osaka Kyoiku University, Kashiwara, Japan

• It serves as a comprehensive text for teaching and learning knot theory from elementary school to high school
• It provides a model for cooperation between mathematicians and mathematics educators based on substantial mathematics
• It is a thorough introduction to the Japanese art of lesson studies again in the context of substantial mathematics

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Table of contents (7 chapters)

Front matter, what is knot theory why is it in mathematics.

• Akio Kawauchi, Tomoko Yanagimoto

The Evolution of Mathematics Education-forwarding the research and practice of teaching knot theory in mathematics education-

The background of developing teaching contents of knot theory, education practice in elementary school, education practices in junior high school, education practices in senior high schools, education practice at the university as liberal arts and teacher education, back matter.

• Arithmetic Education
• Experimental teaching
• Knot Theory
• Mathematics Education

About this book

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Editors and Affiliations

Akio Kawauchi

Tomoko Yanagimoto

About the editors

Bibliographic information.

Book Title : Teaching and Learning of Knot Theory in School Mathematics

Editors : Akio Kawauchi, Tomoko Yanagimoto

DOI : https://doi.org/10.1007/978-4-431-54138-7

Publisher : Springer Tokyo

eBook Packages : Mathematics and Statistics , Mathematics and Statistics (R0)

Copyright Information : Osaka Municipal Universities Press 2012

Hardcover ISBN : 978-4-431-54137-0 Published: 27 June 2012

Softcover ISBN : 978-4-431-56120-0 Published: 23 August 2016

eBook ISBN : 978-4-431-54138-7 Published: 27 June 2012

Edition Number : 1

Number of Pages : XIV, 188

Topics : Geometry , Topology , Mathematics Education

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Using ideas from game theory to improve the reliability of language models

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Imagine you and a friend are playing a game where your goal is to communicate secret messages to each other using only cryptic sentences. Your friend's job is to guess the secret message behind your sentences. Sometimes, you give clues directly, and other times, your friend has to guess the message by asking yes-or-no questions about the clues you've given. The challenge is that both of you want to make sure you're understanding each other correctly and agreeing on the secret message.

MIT Computer Science and Artificial Intelligence Laboratory (CSAIL) researchers have created a similar "game" to help improve how AI understands and generates text. It is known as a “consensus game” and it involves two parts of an AI system — one part tries to generate sentences (like giving clues), and the other part tries to understand and evaluate those sentences (like guessing the secret message).

The researchers discovered that by treating this interaction as a game, where both parts of the AI work together under specific rules to agree on the right message, they could significantly improve the AI's ability to give correct and coherent answers to questions. They tested this new game-like approach on a variety of tasks, such as reading comprehension, solving math problems, and carrying on conversations, and found that it helped the AI perform better across the board.

Traditionally, large language models answer one of two ways: generating answers directly from the model (generative querying) or using the model to score a set of predefined answers (discriminative querying), which can lead to differing and sometimes incompatible results. With the generative approach, "Who is the president of the United States?" might yield a straightforward answer like "Joe Biden." However, a discriminative query could incorrectly dispute this fact when evaluating the same answer, such as "Barack Obama."

So, how do we reconcile mutually incompatible scoring procedures to achieve coherent, efficient predictions? 

"Imagine a new way to help language models understand and generate text, like a game. We've developed a training-free, game-theoretic method that treats the whole process as a complex game of clues and signals, where a generator tries to send the right message to a discriminator using natural language. Instead of chess pieces, they're using words and sentences," says Athul Jacob, an MIT PhD student in electrical engineering and computer science and CSAIL affiliate. "Our way to navigate this game is finding the 'approximate equilibria,' leading to a new decoding algorithm called 'equilibrium ranking.' It's a pretty exciting demonstration of how bringing game-theoretic strategies into the mix can tackle some big challenges in making language models more reliable and consistent."

When tested across many tasks, like reading comprehension, commonsense reasoning, math problem-solving, and dialogue, the team's algorithm consistently improved how well these models performed. Using the ER algorithm with the LLaMA-7B model even outshone the results from much larger models. "Given that they are already competitive, that people have been working on it for a while, but the level of improvements we saw being able to outperform a model that's 10 times the size was a pleasant surprise," says Jacob. 

"Diplomacy," a strategic board game set in pre-World War I Europe, where players negotiate alliances, betray friends, and conquer territories without the use of dice — relying purely on skill, strategy, and interpersonal manipulation — recently had a second coming. In November 2022, computer scientists, including Jacob, developed “Cicero,” an AI agent that achieves human-level capabilities in the mixed-motive seven-player game, which requires the same aforementioned skills, but with natural language. The math behind this partially inspired the Consensus Game. 

While the history of AI agents long predates when OpenAI's software entered the chat in November 2022, it's well documented that they can still cosplay as your well-meaning, yet pathological friend. 

The consensus game system reaches equilibrium as an agreement, ensuring accuracy and fidelity to the model's original insights. To achieve this, the method iteratively adjusts the interactions between the generative and discriminative components until they reach a consensus on an answer that accurately reflects reality and aligns with their initial beliefs. This approach effectively bridges the gap between the two querying methods. 

In practice, implementing the consensus game approach to language model querying, especially for question-answering tasks, does involve significant computational challenges. For example, when using datasets like MMLU, which have thousands of questions and multiple-choice answers, the model must apply the mechanism to each query. Then, it must reach a consensus between the generative and discriminative components for every question and its possible answers. 

The system did struggle with a grade school right of passage: math word problems. It couldn't generate wrong answers, which is a critical component of understanding the process of coming up with the right one. 

“The last few years have seen really impressive progress in both strategic decision-making and language generation from AI systems, but we’re just starting to figure out how to put the two together. Equilibrium ranking is a first step in this direction, but I think there’s a lot we’ll be able to do to scale this up to more complex problems,” says Jacob.   

An avenue of future work involves enhancing the base model by integrating the outputs of the current method. This is particularly promising since it can yield more factual and consistent answers across various tasks, including factuality and open-ended generation. The potential for such a method to significantly improve the base model's performance is high, which could result in more reliable and factual outputs from ChatGPT and similar language models that people use daily. 

"Even though modern language models, such as ChatGPT and Gemini, have led to solving various tasks through chat interfaces, the statistical decoding process that generates a response from such models has remained unchanged for decades," says Google Research Scientist Ahmad Beirami, who was not involved in the work. "The proposal by the MIT researchers is an innovative game-theoretic framework for decoding from language models through solving the equilibrium of a consensus game. The significant performance gains reported in the research paper are promising, opening the door to a potential paradigm shift in language model decoding that may fuel a flurry of new applications."

Jacob wrote the paper with MIT-IBM Watson Lab researcher Yikang Shen and MIT Department of Electrical Engineering and Computer Science assistant professors Gabriele Farina and Jacob Andreas, who is also a CSAIL member. They presented their work at the International Conference on Learning Representations (ICLR) earlier this month, where it was highlighted as a "spotlight paper." The research also received a “best paper award” at the NeurIPS R0-FoMo Workshop in December 2023.

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MIT researchers have developed a new procedure that uses game theory to improve the accuracy and consistency of large language models (LLMs), reports Steve Nadis for Quanta Magazine . “The new work, which uses games to improve AI, stands in contrast to past approaches, which measured an AI program’s success via its mastery of games,” explains Nadis. 

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Evolving market dynamics foster consumer inattention that can lead to risky purchases

Researchers have developed a new theory of how changing market conditions can lead large numbers of otherwise cautious consumers to buy risky products such as subprime mortgages, cryptocurrency or even cosmetic surgery procedures.

These changes can occur in categories of products that are generally low risk when they enter the market. As demand increases, more companies may enter the market and try to attract consumers with lower priced versions of the product that carry more risk. If the negative effects of that risk are not immediately noticeable, the market can evolve to keep consumers ignorant of the risks, said Michelle Barnhart, an associate professor in Oregon State University's College of Business and a co-author of a new paper.

"It's not just the consumer's fault. It's not just the producer's fault. It's not just the regulator's fault. All these things together create this dilemma," Barnhart said. "Understanding how such a situation develops could help consumers, regulators and even producers make better decisions when they are faced with similar circumstances in the future."

The researchers' findings were recently published in the Journal of Consumer Research . The paper's lead author is Lena Pellandini-Simanyi of the University of Lugano in Switzerland.

Barnhart, who studies consumer culture and market systems; has researched credit and debit in the U.S. Pellandini-Simanyi, a sociologist with expertise in consumer markets, has studied personal finance in European contexts. Together they analyzed the case of the Hungarian mortgage crisis to understand how people who generally view themselves as risk averse end up pursuing a high-risk product or service.

To better understand the consumer mindset, the researchers conducted 47 interviews with Hungarian borrowers who took out low-risk mortgages in the local forint currency or in higher risk foreign currency as the Hungarian mortgage industry evolved between 2001 and 2010. They also conducted a larger survey of mortgage borrowers, interviewed 37 finance and mortgage industry experts and financial regulators and analyzed regulatory documents and parliamentary proceedings.

They found patterns that led to mortgages becoming riskier over time and social and marketplace changes that lead consumers to enter into a state of collective ignorance of increasing risks. In addition, they identified characteristics that encouraged these patterns. Other markets with these characteristics are likely to develop in a similar way.

"Typically, when there is a new product on the market, people are quite skeptical. The early adopters carefully examine this product, they become highly educated about it and do a lot of work to determine if the risk is too high," Pellandini-Sumanyi said. "If they deem the risk too high, they don't buy it."

But if those early adopters use the new product or service successfully, the next round of consumers is likely to assume the product will work for them in a similar fashion without examining it in as much detail, even if the quality of the product has been reduced, the researchers noted.

"Then everything starts to spiral, with quality dropping in the rush to meet consumer demand and maintain profits, and consumers relying more and more on social information that suggests this is a safe purchase without investigating how the risks might have changed," Barnhart said.

"It also can lead to a 'prudence paradox,' where the most risk averse people wait to enter the market until the end stages and end up buying super risky products. They exercise caution by waiting but they wait so long, they end up with the worst products."

The spiral is typically only broken through intervention, either through market collapse or regulation. For example, while cosmetic surgery is relatively safe, an increase in availability of inexpensive procedures at facilities that lacked sufficient equipment and expertise led to a rise in botched procedures until regulation caught up.

"These findings demonstrate the power of social information," Barnhart said. "In this environment, it's very difficult for any individual consumer to pay attention to and assess risk because doing so is so far outside of the norm."

To protect themselves against collective ignorance, consumers should ensure that they are weighing their personal risk against others whose experiences are actually similar, Pellandini-Sumanyi said.

"Make sure this is an apples-to-apples comparison of products and the consumers' circumstances," she said.

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• Probability theory

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Materials provided by Oregon State University . Original written by Michelle Klampe. Note: Content may be edited for style and length.

Journal Reference :

• Léna Pellandini-Simányi, Michelle Barnhart. The Market Dynamics of Collective Ignorance and Spiraling Risk . Journal of Consumer Research , 2024; DOI: 10.1093/jcr/ucae018

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