Lickorish - Introduction to Knot Theory (1991).pdf - Abstract Algebra - Linear Algebra - Kuya

Graduate Texts in Mathematics

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W.B. Raymond Lickorish

An Introduction to

Knot Theory

With 114 Illustrations

W.B. Raymond Lickorish

Professor of Geometric Topology, University of Cambridge,

and Fellow of Pembroke College, Cambridge

Department of Pure Mathematics and Mathematical Statistics

Cambridge CB2 ISB

England

Editorial Board

S. Axler

Mathematics Department

San Francisco State

University

San Francisco, CA 94132

USA

F.W. Gehring

Mathematics Department

East Hall

University of Michigan

Ann Arbor, MI 48109

USA

K.A. Ribet

Department of Mathematics

University of California

at Berkeley

Berkeley, CA 94720

USA

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Mathematics Subject Classification (1991): 57-01, 57M25, 16534, 57M05

Preface

'fhis account is an introduction to mathematical knot theory, the theory of knots

and links ofsimple closed curves in three-dimensional space. Knots can be studied

at lTIany levels and from lTIany points of view. They can be admired as artifacts of

the decorative arts and crafts, or viewed as accessible intimations ofa geometrical

sophistication that may never be attained. The study of knots can be given some

ITIotivation 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-

olnetry of three-dimensional space can be explored by knotting phenolnena using

precise Inathematics. The aim will be to find invariants that distinguish knots, to

investigate geolTIetric properties of knots and to see something of the way they

interact with lnore 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.

There are other works on knot theory wri tten at this level; indeed most of them

are listed in the bibliography. However, the quantity of what may reasonably be

tcnned mathematical knot theory has expanded enormously in recent years. Much

of the newly discovered material is not particularly difficult and has a right to be

included in an introduction. This makes some ofthe excellent established treatises

SCCITI a I ittle dated. However, concentrating entirely on developments of the past

decade gives a lTIOSt misleading view of the subject. An attempt is made here to

outline sOlne of the highlights from throughout the twentieth century, with a little

hias towards recent discoveries.

'rhe present size of the subject Incans that a choice of topics must be ITIade for

Illclusion in any first course or book of reasonable length. Such selection nlust hc

suhjective. I\n attcillpt has heen Inade here to give the flavour and the results rrolll

lhree or f()tlr 1l1aln tlThlliqul's and not 10 heroillc unduly l'lllllL'shcd ill allY orthclll.

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