The knot book: an elementary introduction to the mathematical theory of knots.

*(English)*Zbl 0840.57001
New York, NY: Freeman and Co. xiii, 306 p. (1994).

The book under review gives (as the title suggests) an elementary introduction into knot theory: “This book can be and has been used effectively as a textbook in classes. With the exception of a few spots, the book assumes only a familiarity with high school algebra.” (Cited from the preface). Here are the main topics, which are discussed in the book:

Chapter 1 contains the definition of knots, links and their equivalence. Knot projections and the Reidemeister moves are introduced. Most invariants to be discussed in the book are defined combinatorially, i.e. the invariant is first defined in terms of a knot projection and then shown to be invariant under Reidemeister moves. The first examples are the linking number and the tricolorability of a knot (which is used to show nontriviality of the trefoil knot). Chapter 2 discusses the problem of tabulating knots (how can one feed a knot into a computer?). In particular Dowker’s notation for knots and Conway’s tangle notation are explained. It is shown that a knot projection is equivalent to a signed planar graph. Chapter 3: The following knot invariants are examined: unknotting number, bridge number and crossing number. Chapter 4: Surfaces are considered. It is explained that these can be classified by their genus resp. Euler characteristic and their orientability resp. non-orientability. The construction of a Seifert surface of a knot is given and the genus of a knot is defined. It is proved that the genus behaves additively with respect to composition of knots. Chapter 5 is called “Types of knots” and contains sections on torus knots, satellite knots and hyperbolic knots. Moreover the connection between knots and braids is discussed in this chapter. It is proved that any knot can be obtained by closing a braid. Markov’s theorem is cited (but not proved). There is also a section on “almost alternating knots”, i.e. knots that are not alternating but have a projection which can be made alternating by one crossing change. Chapter 6: The Kauffman bracket polynomial (resp. the Jones polynomial) is defined. It is used to show that any two reduced alternating projections of the same knot have the same number of crossings. The Alexander polynomial and the HOMFLY polynomial are introduced via their skein relations. The Kauffman bracket polynomial is used to study amphicheirality. Chapter 7: Some connections between knot theory and natural sciences are explained, in particular: Knottedness of DNA, synthesis of knotted molecules, chirality of molecules, connections with statistical mechanics (state sums and partition functions). Chapter 8: Some relations with graph theory are explained: knots and links in graphs, polynomial invariants of graphs. Chapter 9: 3-manifolds are defined. It is (almost) proved that any (closed) admits a Heegard splitting. Dehn surgeries are explained. Chapter 10 gives some ideas of higher dimensional knotting. Finally there is a section called “knot jokes and pastimes” and an appendix which contains pictures and invariants for all knots up to nine crossings and all two resp. three component links up to eight resp. seven crossings.

Throughout the book there are lots of exercises of various degrees of difficulty. Many “unsolved questions” provide opportunity for further research. I liked reading this book.

Chapter 1 contains the definition of knots, links and their equivalence. Knot projections and the Reidemeister moves are introduced. Most invariants to be discussed in the book are defined combinatorially, i.e. the invariant is first defined in terms of a knot projection and then shown to be invariant under Reidemeister moves. The first examples are the linking number and the tricolorability of a knot (which is used to show nontriviality of the trefoil knot). Chapter 2 discusses the problem of tabulating knots (how can one feed a knot into a computer?). In particular Dowker’s notation for knots and Conway’s tangle notation are explained. It is shown that a knot projection is equivalent to a signed planar graph. Chapter 3: The following knot invariants are examined: unknotting number, bridge number and crossing number. Chapter 4: Surfaces are considered. It is explained that these can be classified by their genus resp. Euler characteristic and their orientability resp. non-orientability. The construction of a Seifert surface of a knot is given and the genus of a knot is defined. It is proved that the genus behaves additively with respect to composition of knots. Chapter 5 is called “Types of knots” and contains sections on torus knots, satellite knots and hyperbolic knots. Moreover the connection between knots and braids is discussed in this chapter. It is proved that any knot can be obtained by closing a braid. Markov’s theorem is cited (but not proved). There is also a section on “almost alternating knots”, i.e. knots that are not alternating but have a projection which can be made alternating by one crossing change. Chapter 6: The Kauffman bracket polynomial (resp. the Jones polynomial) is defined. It is used to show that any two reduced alternating projections of the same knot have the same number of crossings. The Alexander polynomial and the HOMFLY polynomial are introduced via their skein relations. The Kauffman bracket polynomial is used to study amphicheirality. Chapter 7: Some connections between knot theory and natural sciences are explained, in particular: Knottedness of DNA, synthesis of knotted molecules, chirality of molecules, connections with statistical mechanics (state sums and partition functions). Chapter 8: Some relations with graph theory are explained: knots and links in graphs, polynomial invariants of graphs. Chapter 9: 3-manifolds are defined. It is (almost) proved that any (closed) admits a Heegard splitting. Dehn surgeries are explained. Chapter 10 gives some ideas of higher dimensional knotting. Finally there is a section called “knot jokes and pastimes” and an appendix which contains pictures and invariants for all knots up to nine crossings and all two resp. three component links up to eight resp. seven crossings.

Throughout the book there are lots of exercises of various degrees of difficulty. Many “unsolved questions” provide opportunity for further research. I liked reading this book.

Reviewer: H.M.UnsĂ¶ld (Berlin)

##### MSC:

57-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |