Explorations in topology. Map coloring, surfaces and knots (Q2865995)

The review of the first edition can be found at [\textit{D. Gay}, Explorations in topology. Map coloring, surfaces and knots. Amsterdam: Elsevier/Academic Press (2007; Zbl 1111.57001)]. The second edition is virtually identical to the first edition and the reader can either continue reading this review or look at the review of the first edition.NEWLINENEWLINEThere are several books that introduce topics in low dimensional topology with very few (or no formal) prerequisites. The book reviewed here is different from all of these by its unique style: The author tries to introduce the reader to topological thinking by creating a narrative of an imaginary company called Acme Maps. The employees of this company embark on a mathematical journey through their work starting with the coloring of maps, leading through a classification of surfaces, continuing with the coloring problem on surfaces, and ending with an introduction of knot theory by studying colorings of knot diagrams. The topics are often progressing by a fictive discussion among the employees of Acme Maps. In summary, the book is not written in the usual style of mathematical texts based on lemmas, theorems and proofs. Many books have exercises at the end of each section. Here instead of traditional exercises there are frequent ``Your Turn'' and ``Investigation'' sections popping up in the middle of a chapter, where the students are asked to develop a mathematical example or idea that has arisen in the narrative. Typical problems ask the students to look at more examples, make paper models, or simply ask students to investigate issues that the employees of Acme Maps need to resolve. There are hundreds and hundreds of well designed figures that make this book very visually attractive. The book is suitable for undergraduates or even excellent high school students that have a willingness and energy to invest time in the (often open ended) questions posted by the text. Despite having no formal prerequisites the tasks that are asked of the reader are challenging and require clear thinking. This text could be an exiting tool for self study or a non-traditional course that is not just based on lectures.