Knot Theory

Carus Mathematical Monographs #24

by Charles Livingston

Knot Theory, a lively exposition of the mathematics of knotting, will appeal to a diverse audience from the undergraduate seeking experience outside the traditional range of studies to mathematicians wanting a leisurely introduction to the subject. Graduate students beginning a program of advanced study will find a worthwhile overview, and the reader will need no training beyond linear algebra to understand the mathematics presented. The interplay between topology and algebra, known as algebraic topology, arises early in the book, when tools from linear algebra and from basic group theory are introduced to study the properties of knots, including one of mathematics' most beautiful topics, symmetry. The book closes with a discussion of high-dimensional knot theory and a presentation of some of the recent advances in the subject - the Conway, Jones and Kauffman polynomials. A supplementary section presents the fundamental group, which is a centerpiece of algebraic topology.

Genres: Mathematics

258 pages, Hardcover

First published January 1, 1993

Reviews

[Justpassingby]

The book starts with elementary definitions of (piecewise linear) knots and knot diagrams, a statement of the general equivalence problem and an elementary treatment of Reidemeister moves. Subsequent chapters treat the combinatorial (mod p labeling, Alexander polynomial), geometric (Seifert surfaces) and algebraic (group labeling) perspectives separately, and then introduce combinations of techniques and relationships among them.

Some of the choices that the author made contribute to making this a very accessible introduction to the subject.

One fortunate choice is to gradually introduce more and more sophisticated invariants, rather than presenting a simpler invariant as a special case of the more complex one. Colouring a knot using 3 colours is "just" labeling it with residue classes modulo 3, but labeling by residue classes modulo any prime p becomes a lot more acceptable and understandable once you have tried to colour a few knot diagrams using 3 colours. Mod p labeling in its turn makes the determinant, the Alexander polynomial and group presentations more palatable.

Another smart choice is to offer a balanced mix of elementary proofs and statements of results without proof. This keeps the pace going without actually skimming over the surface.

Group theory is not taken for granted; in particular, there is a good explanation of the general symmetry group as well as the concept of generators and presentations. Similarly, we get elementary concepts from the classification of surfaces as well as the definition of the fundamental group.

Finally but not unimportantly, the exercises are doable. It is clear that the author has taken the trouble to work them out by himself, to make sure that not a single exercise risks discouraging the reader (even if, or perhaps precisely for this reason, no solutions are offered at the end).

[Barb]

had to return it to the library