This book, the 36th in the Knots and Everything series, ably illustrates the wide range of research involving knots. It provides a rich path through the four ‘countries’ of the knot world: physics, life sciences, numerical simulations and mathematics. There are different countries for various tastes but all are visited with the theory of knots as a companion. It can thus be seen that particle physics returns to knots – as Thomson did by the end of the 19th century with his atomic model – which can be associated with electric flux tubes used for classifying some hadronic states. In plasma physics, it appears that knotted magnetic flux tubes are much more stable than unknotted single loops. Physics is also involved in modelling polymers using knot energies. But it is in life sciences that most of the applications in this book are proposed. DNA knots and protein folds are discussed using various approaches such as Monte-Carlo simulations, microscopy, topological concepts and thermodynamics. One of the most amazing examples is illustrated by the pictures of the uncommon formation of incredibly knotted umbilical cords. Using atomic force microscopy, DNA knots can be observed. It is also known that more complex (i.e. more compact) knots migrate in a gel faster than simple knots, thus explaining why gel electrophoresis can separate knotted DNA according to knot type. On the other hand, knot type can be used to measure the effect of chemical composition on polymer shape. In fact, many applications are devoted to polymers: topological constraints can be related to anomalous osmotic pressure, entanglements to elastic properties, etc. Polymers appears as one of the preferred subjects for stimulating numerical simulations of knot configurations and developing algorithms to investigate properties such as ropelength, i.e. the quotient of the knot length by its thickness.

The book is stimulating because it provides many different types of analysis; analytic computations, numerical simulations and new concepts are discussed with different backgrounds. The diversity of the domains covered occasionally makes the book quite difficult since there are undoubtedly fields that the reader will encounter in which they are not specialized. For instance, there are large gaps between quantum chromodynamics, topological constraints, minimal flat knotted ribbons and gel-electrophoresis. Once this difficulty is left behind (it is always possible to consult the bibliography to obtain more details), the main interest of the book is to provide a wide range of approaches to investigating knots and, consequently, to familiarize the reader with topological invariants, numerical techniques and other statistical techniques according to their own taste.

Since it is devoted to a large number of approaches and applications, this book is recommended to anyone interested in using knots in applied science. Many techniques and concepts are discussed and there is likely to be one for opening a “breach” in the reader’s problem. Among others, there is an interesting open problem concerning the application of knot theory to open strings. Such investigations could have many implications in applied science where closed loops are often hard to identify. The book already provides stimulating approaches to this problem, leading to the concept of knottedness and minimal flat knotted ribbons. It is definitely not an introductory book but provides a nice opportunity to be introduced to advanced research and applications in knot theory. One of its main interests is the intermingling of different research areas such as solid state physics, statistical physics and biophysics. Polymer chains are simply disordered knots that can be viewed on a lattice investigated using statistical properties. Thus thermodynamics and topology can be used together to help understand the complexity of underlying knots and links. What are the possible knots matching a given constraint? Numerical methods can be used to produce various knot types and compute some properties (e.g. ropelength, topological entropic force, average crossing number and probability of knotting). It is interesting to see that crossing numbers can be related to complexity measure and that knots can also inspire statistical approaches. In order to do that, special attention is paid to knots with a large number of crossings. The book offers a nice bridge between various topics too often considered as isolated islands.