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An Introduction to Knot Theory (Graduate Texts in Mathematics), by W.B.Raymond Lickorish
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A selection of topics which graduate students have found to be a successful introduction to the field, employing three distinct techniques: geometric topology manoeuvres, combinatorics, and algebraic topology. Each topic is developed until significant results are achieved and each chapter ends with exercises and brief accounts of the latest research. What may reasonably be referred to as knot theory has expanded enormously over the last decade and, while the author describes important discoveries throughout the twentieth century, the latest discoveries such as quantum invariants of 3-manifolds as well as generalisations and applications of the Jones polynomial are also included, presented in an easily intelligible style. Readers are assumed to have knowledge of the basic ideas of the fundamental group and simple homology theory, although explanations throughout the text are numerous and well-done. Written by an internationally known expert in the field, this will appeal to graduate students, mathematicians and physicists with a mathematical background wishing to gain new insights in this area.
- Sales Rank: #1364205 in Books
- Published on: 1997-10-03
- Original language: English
- Number of items: 1
- Dimensions: 9.21" h x .63" w x 6.14" l, .96 pounds
- Binding: Hardcover
- 204 pages
Review
W.B.R. Lickorish
An Introduction to Knot Theory
"This essential introduction to vital areas of mathematics with connections to physics, while intended for graduate students, should fall within the ken of motivated upper-division undergraduates."―CHOICE
From the Back Cover
This volume is an introduction to mathematical knot theory - the theory of knots and links of simple closed curves in three-dimensional space. It consists of a selection of topics that graduate students have found to be a successful introduction to the field. Three distinct techniques are employed: geometric topology manoeuvres; combinatorics; and algebraic topology. Each topic is developed until significant results are achieved, and chapters end with exercises and brief accounts of state-of-the-art research. What may reasonably be referred to as knot theory has expanded enormously over the last decade, and while the author describes important discoveries from throughout the twentieth century, the latest discoveries such as quantum invariants of 3-manifolds - as well as generalisations and applications of the Jones polynomial - are also included, presented in an easily understandable style. Thus, this constitutes a comprehensive introduction to the field, presenting modern developments in the context of classical material. Readers are assumed to have knowledge of the basic ideas of the fundamental group and simple homology theory, although explanations throughout the text are plentiful and well done. Written by an internationally known expert in the field, this volume will appeal to graduate students, mathematicians, and physicists with a mathematical background who wish to gain new insights in this area.
Most helpful customer reviews
14 of 14 people found the following review helpful.
Thorough and intriguing
By J. Conant
The level of detail in this book is just right. It provides complete and rigorous proofs without getting bogged down in too much detail. In addition to some other knot theory standards, it has an excellent section on 3-manifold invariants arising from the Temperley-Lieb algebra. This book is appropriate for those who have a knowledge of algebraic topology.
1 of 1 people found the following review helpful.
Best book out there for Knot Theory, go for it!
By Ricardo Avila
I had always felt the need to understand knot theory or at least to have an introductory knowledge of it. Knots, ever since I met them have intrigue me and fascinate as how those weird entangled pieces of string, i.e. in other words their funny and beautiful shapes, can be formulated in a quantitative way through the use of mathematics, specially the topological invariants of polynomials were my concern. They were a real mystery to me as I suppose Quantum Mechanics was a mystery to me when I was at high school. Well, the story is that I had tried several books to dive into knot theory but it was complete failure until I pick this book. I must say though, that when I did it I already had a knowledge of Algebraic Topology which is needed. Before saying anything else I must tell you that I am a physicist and so I have not try to understand every proof to every proposition, lemma and theorem but just to read it as a novel learning facts here and there. I also, as a physicist had a special interest in the Jones Polynomial since it pops up in topological quantum field theory thanks to the work of Edward Witten. So, what can I say? I only read the first 6 chapters (It has 16 in total) and I'm satisfy because I did encounter and comprehend the Jones polynomial and also the Alexander polynomial. Chapter 1 "A Beginning for Knot Theory" is very nice, it gives you a general flabour and taste of different elements that are used in knot theory like the definition of the Redemeister moves, what is a Link, the linking number, prime knots, pretzel knots, conway characterization of a knot through continued fractions etc; Chapter 2 "Seifert Surfaces and Knot Factorization" essentially introduces the notion of a Seifert surface associated to a knot which is basically a connected oriented surface which has the knot as its boundary, Chapter 3 "The Jones Polynomial" introduces to the reader this polynomial which is a topological invariant constructed out of the 'bracket polynomial' (which is not an invariant) and the "writhe" of the knot, which is essentially a number that has to do with the over and under crosses of the knot (basically the sum of them); Chapter 4 "Geometry of Alternating Links" introduces notions associated to alternating knots; Chapter 5 "The Jones Polynomial of an Alternating Link" gives some facts about alternating links and the J.P. and finally (the last chapter I read) Chapter 6 "The Alexander Polynomial" introduces this other (older than Jones) topological invariant of a knot which is defined as the determinant of certain complicated matrix associated to the number of crossing of the knot which are found when transversing a curve which is a generator (in the sense of Homology) of the Seifert surface associated to the given knot, I would gladly had keep on reading further but I realize that I had to come back to physics (Quantum Field Theory, General Relativity and Quantum Gravity) and also to religion and my spiritual growth so having encountered the Jones and Alexander polynomials I felt satisfied and I decided to put it back on my shelf for a while (whether it will be short or long I don't know), but it is a nice written text so my suggestion for you is: go for it you won't regret it!
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