A knot is a smooth curve drawn in three dimensional space that begins and ends at the same place. A theorem of K. Reidemeister states that two knots are equivalent if and only if they have diagrams that may be obtained from each other by a finite sequence of three diagrammatic moves. If you have two knots and you are unable to find a sequence of Reidemeister moves connecting them, you might begin to suspect that they are not equivalent after all. But how can it be proved for certain? The fascinating text of M. Elhamdadi and S. Nelson gives a practical introduction to a powerful algebraic technique called quandles.
A quandle is a set \(X\) together with a binary operation \(\triangleright:X \times X \to X\) satisfying (I) \(x \triangleright x=x\) for all \(x \in X\), (II) for all \(y \in X\), the map \(\beta_y(x)=x \triangleright y\) is invertible, and (III) for all \(x,y,z \in X\), \((x \triangleright y) \triangleright z=(x \triangleright z) \triangleright (y \triangleright z)\). The criteria (I), (II), (III) correspond naturally to the Reidemeister moves for oriented knots. To each diagram we may associate a fundamental quandle as follows. The arcs in the diagram are labeled with distinct letters. At each crossing we have a relation. The fundamental quandle is the quandle generated by the arc letters, the crossing relations, and the relations (I), (II), (III).
In Chapter 2, the book gently introduces beginners to the abstract algebra needed to understand and manipulate quandles. Prerequisites are minimal. A reader comfortable with elementary set theory, modular arithmetic, and computational linear algebra will have no trouble working through the entirety of the text. All topology and knot theory needed for success is contained within Chapter 1. From these basic principles the authors are able to reach deep into the theory. For example, quandle cocycle enhancements are an important tool in quandle theory. While it would be tempting to leave this to an advanced course, the authors instead provide an excellent preparatory section on cohomology. This allows for later discussion (Chapter 6) of cocycle enhancements and some of their geometric applications (e.g. detecting non-invertible spheres in four dimensional space and obstructions to tangle embeddings).
The heart of the text lies in Chapters 3, 4, and 5. These chapters define quandles and explain how to use them to study knots. The reader begins working with kei, which are the simplest relative of quandles. Quandles, racks, bikei, biquandles, and biracks are introduced in turn. Examples are used to illustrate why all of these structures are necessary. As the structures increase in complexity so also does their power to distinguish knots, links, and framed links. The computational complexity of course also increases, but this does not detract from their utility. The numerous examples and exercises in the text can all be reasonably computed by hand.
Chapter 4 motivates the study of quandles by relating their definition to other important algebraic structures in knot theory: knot groups, braid groups, and Alexander modules. The fundamental quandle of a knot can be interpreted geometrically as the set of homotopy classes of paths from the fixed basepoint to the boundary torus of the knot complement. This is similar to the definition to the knot group, i.e. the fundamental group of the knot complement. The fundamental quandle of a knot is moreover a complete knot invariant up to reflection. This means that if two oriented knots have the same fundamental quandle, then they are either ambient isotopic or mirror images of one another (with reveresed orienation). Quandles thus elegantly encode knots into algebra.
Knot theory in general studies more than just curves drawn in three dimensional space. There is a large literature on knotted embeddings of surfaces in four dimensional space. Virtual knots, which are knots in thickened surfaces considered up to “stabilization”, are another active area of research. Quandles can be used to study these generalized knot theories as well. The last chapter of the text consists of a series of essays outlining applications to tangles, knotted surfaces, and virtual knots. The interested reader could move effortlessly from here to a more detailed treatment, such as [Surfaces in 4-space. Encyclopaedia of Mathematical Sciences 142. Low-Dimensional Topology 3. Berlin: Springer. xiv, 213 p. (2004; Zbl 1078.57001)] by S. Carter, S. Kamada, and M. Saito.
This book will appeal to anyone interested in algebra or topology. It would be well suited for a textbook for an undergraduate course or independent study in knot theory. There is plenty inside for both the beginner and expert to enjoy!