Knots and Links - Assetsassets.cambridge.org/97805215/48311/frontmatter/9780521548311... · Knots and Links Knotsandlinksarestudiedbymathematicians,andarealsoﬁndingincreasing applicationinchemistryandbiology.Manynaturallyoccurringquestionsareoften

Knots and Links

Knots and links are studied by mathematicians, and are also finding increasingapplication in chemistry and biology. Many naturally occurring questions are oftensimple to state, yet finding the answers may require ideas from the forefront ofresearch.

This readable and richly illustrated book explores selected topics in depth in away that makes contemporary mathematics accessible to an undergraduate audi-ence. It can be used for upper-division courses, and assumes only knowledge ofbasic algebra and elementary topology. The techniques developed include combi-natorics applied to diagrams, the study of surface intersections and ‘cut and paste’surgery, skein theory of polynomial invariants, and the properties of tangles. To-gether with standard topics such as Seifert matrices and Alexander and Jonespolynomials, the book explains polygonal and smooth presentations, the surgeryequivalence of surfaces, the behaviour of invariants under factorisation and thesatellite construction, the arithmetic of Conway’s rational tangles, and arc presen-tations. The families of torus knots, pretzel knots, rational (or 2-bridge) links anddoubles of the trefoil recur as examples throughout the text.

Alongside the systematic development of the main theory, there are discussionsections that cover historical aspects, motivation, possible extensions and applica-tions. Many examples and exercises are included to show both the power and thelimitations of the techniques developed.

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The construction of a braided bidirectional satellite. The vertical line is an axis that provides the reference framefor the structure. The transparent tube is a torus knotted in the form of a trefoil and is the companion torus of thesatellite. Its arrangement is based on an arc presentation of the companion knot: an assembly of five semicirclesmeeting at the axis (see Figure 10.6). The two loops inside the torus form the satellite link. They run around thetorus in opposite directions (the satellite is bidirectional), but they run around the axis in the same direction (sothey are braided). (Image created with PovRay.)






Knots and Links

Peter R. Cromwell






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© Cambridge University Press 2004

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First published 2004

A catalogue record for this publication is available from the British Library

Library of Congress Cataloguing in Publication dataCromwell, Peter R., 1964–Knots and links / Peter R. Cromwell.p. cm.Includes bibliographical references and index.ISBN 0 521 83947 5 (hardback) – ISBN 0 521 54831 4 (paperback)1. Knot theory. 2. Link theory. I. TitleQA612.2.C75 2004514.2242 – dc22 2004045914

isbn 978-0-521-83947-1 Hardbackisbn 978-0-521-54831-1 Paperback

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Contents































Preface






Philosophy

Content






Approach






Prerequisites

Acknowledgements






Notation

Set notation

Standard spaces

Algebraic structures






Links

Link invariants






Others

References






Documents

Knots and Links - Assetsassets.cambridge.org/97805215/48311/frontmatter/9780521548311... · Knots and Links Knotsandlinksarestudiedbymathematicians,andarealsoﬁndingincreasing applicationinchemistryandbiology.Manynaturallyoccurringquestionsareoften