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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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Mathematical Association of America P: (800) 331-1622 F: (240) 396-5647 Email: [email protected]

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• © 1997

An Introduction to Knot Theory

• W. B. Raymond Lickorish 0

Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, and Fellow of Pembroke College,Cambridge, Cambridge, England

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• Written by an internationally acknowledged expert in the field who has won prizes for both exposition and research * Gives a comprehensive introduction to the field, presenting modern developments in the context of classical material * Will appeal to graduate students, mathematicians and physicists with a mathematical background who wish to gain new insights in this area

Part of the book series: Graduate Texts in Mathematics (GTM, volume 175)

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Table of contents (16 chapters)

Front matter, a beginning for knot theory.

W. B. Raymond Lickorish

Seifert Surfaces and Knot Factorisation

The jones polynomial, geometry of alternating links, the jones polynomial of an alternating link, the alexander polynomial, covering spaces, the conway polynomial, signatures and slice knots, cyclic branched covers and the goeritz matrix, the arf invariant and the jones polynomial, the fundamental group, obtaining 3-manifolds by surgery on s 3, 3-manifold invariants from the jones polynomial, methods for calculating quantum invariants, generalisations of the jones polynomial, exploring the homfly and kauffman polynomials, back matter.

• Knot theory
• quantum invariant

Book Title : An Introduction to Knot Theory

Authors : W. B. Raymond Lickorish

Series Title : Graduate Texts in Mathematics

DOI : https://doi.org/10.1007/978-1-4612-0691-0

Publisher : Springer New York, NY

eBook Packages : Springer Book Archive

Copyright Information : Springer Science+Business Media New York 1997

Hardcover ISBN : 978-0-387-98254-0 Published: 03 October 1997

Softcover ISBN : 978-1-4612-6869-7 Published: 12 October 2012

eBook ISBN : 978-1-4612-0691-0 Published: 06 December 2012

Series ISSN : 0072-5285

Series E-ISSN : 2197-5612

Edition Number : 1

Number of Pages : X, 204

Topics : Manifolds and Cell Complexes (incl. Diff.Topology) , Group Theory and Generalizations , Theoretical, Mathematical and Computational Physics

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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!

Parkinson's Disease: New theory on the disease's origins and spread

The nose or the gut? For the past two decades, the scientific community has debated the wellspring of the toxic proteins at the source of Parkinson's disease. In 2003, a German pathologist, Heiko Braak, MD, first proposed that the disease begins outside the brain. More recently, Per Borghammer, MD, with Aarhus University Hospital in Denmark, and his colleagues argue that the disease is the result of processes that start in either the brain's smell center (brain-first) or the body's intestinal tract (body-first).

A new hypothesis paper appearing in the Journal of Parkinson's Disease on World Parkinson's Day unites the brain- and body-first models with some of the likely causes of the disease-environmental toxicants that are either inhaled or ingested. The authors of the new study, who include Borghammer, argue that inhalation of certain pesticides, common dry cleaning chemicals, and air pollution predispose to a brain-first model of the disease. Other ingested toxicants, such as tainted food and contaminated drinking water, lead to body-first model of the disease.

"In both the brain-first and body-first scenarios the pathology arises in structures in the body closely connected to the outside world," said Ray Dorsey, MD, a professor of Neurology at the University of Rochester Medical Center and co-author of the piece. "Here we propose that Parkinson's is a systemic disease and that its initial roots likely begin in the nose and in the gut and are tied to environmental factors increasingly recognized as major contributors, if not causes, of the disease. This further reinforces the idea that Parkinson's, the world's fastest growing brain disease, may be fueled by toxicants and is therefore largely preventable."

Different pathways to the brain, different forms of disease

A misfolded protein called alpha-synuclein has been in scientists' sights for the last 25 years as one of the driving forces behind Parkinson's. Over time, the protein accumulates in the brain in clumps, called Lewy bodies, and causes progressive dysfunction and death of many types of nerve cells, including those in the dopamine-producing regions of the brain that control motor function. When first proposed, Braak thought that an unidentified pathogen, such as a virus, may be responsible for the disease.

The new piece argues that toxins encountered in the environment, specifically the dry cleaning and degreasing chemicals trichloroethylene (TCE) and perchloroethylene (PCE), the weed killer paraquat, and air pollution, could be common causes for the formation of toxic alpha-synuclein. TCE and PCE contaminates thousands of former industrial, commercial, and military sites, most notably the Marine Corps base Camp Lejeune, and paraquat is one of the most widely used herbicides in the US, despite being banned for safety concerns in more than 30 countries, including the European Union and China. Air pollution was at toxic levels in nineteenth century London when James Parkinson, whose 269th birthday is celebrated today, first described the condition.

The nose and the gut are lined with a soft permeable tissue, and both have well established connections to the brain. In the brain-first model, the chemicals are inhaled and may enter the brain via the nerve responsible for smell. From the brain's smell center, alpha-synuclein spreads to other parts of the brain principally on one side, including regions with concentrations of dopamine-producing neurons. The death of these cells is a hallmark of Parkinson's disease. The disease may cause asymmetric tremor and slowness in movement and, a slower rate of progression after diagnosis, and only much later, significant cognitive impairment or dementia.

When ingested, the chemicals pass through the lining of the gastrointestinal tract. Initial alpha-synuclein pathology may begin in the gut's own nervous system from where it can spread to both sides of the brain and spinal cord. This body-first pathway is often associated with Lewy body dementia, a disease in the same family as Parkinson's, which is characterized by early constipation and sleep disturbance, followed by more symmetric slowing in movements and earlier dementia, as the disease spreads through both brain hemispheres.

New models to understand and study brain diseases

"These environmental toxicants are widespread and not everyone has Parkinson's disease," said Dorsey. "The timing, dose, and duration of exposure and interactions with genetic and other environmental factors are probably key to determining who ultimately develops Parkinson's. In most instances, these exposures likely occurred years or decades before symptoms develop."

Pointing to a growing body of research linking environmental exposure to Parkinson's disease, the authors believe the new models may enable the scientific community to connect specific exposures to specific forms of the disease. This effort will be aided by increasing public awareness of the adverse health effects of many chemicals in our environment. The authors conclude that their hypothesis "may explain many of the mysteries of Parkinson's disease and open the door toward the ultimate goal-prevention."

In addition to Parkinson's, these models of environmental exposure may advance understanding of how toxicants contribute to other brain disorders, including autism in children, ALS in adults, and Alzheimer's in seniors. Dorsey and his colleagues at the University of Rochester have organized a symposium on the Brain and the Environment in Washington, DC, on May 20 that will examine the role toxicants in our food, water, and air are playing in all these brain diseases.

Additional authors of the hypothesis paper include Briana De Miranda, PhD, with the University of Alabama at Birmingham, and Jacob Horsager, MD, PhD, with Aarhus University Hospital in Denmark.

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Story Source:

Materials provided by University of Rochester Medical Center . Original written by Mark Michaud. Note: Content may be edited for style and length.

Journal Reference :

• E. Ray Dorsey, Briana R. De Miranda, Jacob Horsager, Per Borghammer. The Body, the Brain, the Environment, and Parkinson’s Disease . Journal of Parkinson's Disease , 2024; 1 DOI: 10.3233/JPD-240019

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6 Common Leadership Styles — and How to Decide Which to Use When

• Rebecca Knight

Being a great leader means recognizing that different circumstances call for different approaches.

Research suggests that the most effective leaders adapt their style to different circumstances — be it a change in setting, a shift in organizational dynamics, or a turn in the business cycle. But what if you feel like you’re not equipped to take on a new and different leadership style — let alone more than one? In this article, the author outlines the six leadership styles Daniel Goleman first introduced in his 2000 HBR article, “Leadership That Gets Results,” and explains when to use each one. The good news is that personality is not destiny. Even if you’re naturally introverted or you tend to be driven by data and analysis rather than emotion, you can still learn how to adapt different leadership styles to organize, motivate, and direct your team.

Much has been written about common leadership styles and how to identify the right style for you, whether it’s transactional or transformational, bureaucratic or laissez-faire. But according to Daniel Goleman, a psychologist best known for his work on emotional intelligence, “Being a great leader means recognizing that different circumstances may call for different approaches.”

• RK Rebecca Knight is a journalist who writes about all things related to the changing nature of careers and the workplace. Her essays and reported stories have been featured in The Boston Globe, Business Insider, The New York Times, BBC, and The Christian Science Monitor. She was shortlisted as a Reuters Institute Fellow at Oxford University in 2023. Earlier in her career, she spent a decade as an editor and reporter at the Financial Times in New York, London, and Boston.

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