The textbook Knots, Links and Their Invariants: An Elementary Course in Contemporary Knot Theory by A. B. Sossinsky provides a rigorous, broad, and engaging view of the mathematical landscape of knots and links for a novice student. This textbook highlights many mathematical results that were originally studied using complex tools that are not within reach of a developing mathematician. In the past 40 years many new insights have been developed that result in approachable pathways to understand some of the cutting-edge results in knot theory. Sossinsky presents these ideas, such as defining the Jones polynomial in terms of the Kauffman bracket, using Conway’s axioms to define the Alexander-Conway polynomial, and introducing the Vassiliev invariants axiomatically, with historical context as well as clear and direct mathematical arguments that rising mathematicians will appreciate. The prerequisites for students are only “a few basic facts of elementary Euclidean geometry, or 2-dimensional and 3-dimensional topology, and of group theory” and the first 11 lectures are relatively self-contained, except for a few references to external knot tables.

The book is organized into 13 lectures, one per week of class, and supporting exercises. Lectures 1-4 include classical topics ranging from Reidemeister’s Theorem to the existence and uniqueness of the Conway polynomial. Lectures 5-7 discuss Kauffman’s bracket, the Jones polynomial, and Alexander’s Theorem on braids. A distinguishing feature of this text are Lectures 8-11 on Vassiliev invariants and the Kontsevich Integrals. Lectures 8 and 9 present Vassiliev invariants using a nicely motivated, clear, and accessible path. Lecture 10 introduces the graded algebra of chord diagrams and Vassiliev-Kontsevich Theorem. Lecture 11 on the Kontsevich Integrals is quite dense, and the difficulty level here takes a sharp turn upwards. This lecture explains the Kontsevich Integral for m=2 by way of two calculations, the first is for a strictly Morse trefoil and second for an unknot with a ’hump.’ The second calculation leads to the definition of the modified Kontsevich integral, the true knot invariant. Although the climb is steep and details are not always included in the final few lectures, the compelling storyline gives students a solid road map towards deeper understanding. The final two Lectures 12 and 13 are a compendium of current areas of research and connections to other fields, and a brief history of knot theory.

The author provides opportunities for students to engage deeply with the topics through exercises at the end of each Lecture. Lectures 1-5 are followed by exercises asking learners to calculate, construct, find, e xpress, s how and check, as well as a few proof problems that are both interesting and accessible to budding mathematicians. In Lectures 6-11, calculations remain a majority of
the exercises, and several more advanced proof problems are included as well. The more challenging problems may require instructors to provide students with
a nudge or two.

Inga Johnson is Professor of Mathematics at Willamette University