An Introduction to Knot Theory

Graduate Texts in Mathematics #175

by W.B. Raymond Lickorish

This account is an introduction to mathematical knot theory, the theory of knots and links of simple closed curves in three-dimensional space. Knots can be studied at many levels and from many points of view. They can be admired as artifacts of the decorative arts and crafts, or viewed as accessible intimations of a geometrical sophistication that may never be attained. The study of knots can be given some motivation in terms of applications in molecular biology or by reference to paral- lels in equilibrium statistical mechanics or quantum field theory. Here, however, knot theory is considered as part of geometric topology. Motivation for such a topological study of knots is meant to come from a curiosity to know how the ge- ometry of three-dimensional space can be explored by knotting phenomena using precise mathematics. The aim will be to find invariants that distinguish knots, to investigate geometric properties of knots and to see something of the way they interact with more adventurous three-dimensional topology. The book is based on an expanded version of notes for a course for recent graduates in mathematics given at the University of Cambridge; it is intended for others with a similar level of mathematical understanding. In particular, a knowledge of the very basic ideas of the fundamental group and of a simple homology theory is assumed; it is, after all, more important to know about those topics than about the intricacies of knot theory.

Genres: Mathematics, Reference

214 pages, Paperback

First published October 3, 1997

Reviews

[Saman · Rating 3 out of 5]

As the name suggests it is an introductory book (in graduate level) about knots. By knot we mean a smooth embedding of a circle in 3 dimensional space. We are interested to know if two different knots are isotopic or not (notion of equivalency), and also we are interested in topological aspects of knots, We solve such problems mostly using invariants.
Knot theory is a very important part of low dimensional topology and the study of 3 manifolds (And recently in some areas of theoretical physics).
The book covers classical invariants in knot theory like Alexander polynomial and also more modern objects like Jones and Homfly polynomials (but not homological invariants like Khovanov Homology).
The book has topological taste, full of geometric deductions and also it has lots of good problems to solve. Some chapters are even appropriate for representing to high school students and some chapters are fairly hard and advanced.