The mystery of knots. Computer programming for knot tabulation (Q2784325)

This book is meant for advanced undergraduates or beginning graduate students who are interested in learning knot theory. The aim of the book is to introduce an algorithm whose output is a knot table up to a given crossing number \(N\). To help the reader understand the algorithm, an introduction to knot theory is given in the first two thirds of the book. The style of the book is very informal aimed at an intuitive understanding of the selected aspects of knot theory that are needed in the algorithm. Many proofs of theorems stated in the text are either not given or only outlined. NEWLINENEWLINENEWLINEThe elements of the algorithm are as follows: To represent a knot diagram for computation the Dowker notation is studied. Using this notation the set of all drawable knot diagrams with a fixed crossing number \(n\) is created for all \(n\leq N\). To distinguish the knots in these sets, the HOMFLYPT-polynomial is computed. Knots with identical HOMFLYPT-polynomials can be distinguished by finding a Wirtinger-presentation of the knot group. From this one can compute different representations into small symmetric groups \(S_{i}\) to try to distinguish different knots. These methods are enough to distinguish all 2977 knots up to (including) \(N=12\) crossings. NEWLINENEWLINENEWLINEThe last third of the book contains a table of all knots up to 12 crossings, including their Dowker notation and HOMFLYPT-polynomial. Additional tables show the knots with identical HOMFLYPT-polynomials and how they are distinguished. It should be pointed out that much more extensive knot tables exist, created using similar methods [\textit{J. Hoste, M. Thistlethwaite}, and \textit{J. Weeks}, Math. Intell. 20, No. 4, 33-48 (1998; Zbl 0916.57008)].NEWLINENEWLINENEWLINEThe only topics of knot theory covered in this book are those that will be needed in the algorithm. In addition, the theorems and definitions are not separated from the body of the text and thus are hard to find for reference purposes. Because of this, this book does not give same quality of general introduction to knot theory as the books of \textit{C. Livingston} [Knot theory (1993; Zbl 0887.57008)] or \textit{C. C. Adams} [The knot book: an elementary introduction to the mathematical theory of knots (1994; Zbl 0840.57001)], which are also aimed at the undergraduate level.