An introduction to quantum and Vassiliev knot invariants
This text is a comprehensive and well written introduction to quantum and Vassiliev invariants of knots. The authors' focus is on interplay between the algebraic and combinatorial structures. The text is laid out for either a one or two semester sequence on this topic. There is sufficient detail for students and exercises. The text is also an excellent reference for researchers interested in quantum and Vassiliev invariants. The book's contents include an introduction to knots and framed knots in Chapters 1--4. The introduction includes Reidemeister's theorem, the Jones polynomial and skein relations, and framed links. The focus then shifts to braids and braid equivalence (the Markov moves). The topic of braids leads naturally into the second section of the text which focuses on quantum invariants and the R-matrix representations of braids. The Yang-Baxter equation is introduced and used construct knot invariants using the trace map applied to braids. Within this framework, the authors construct the Jones and HOMFLYPT polynomials. Calculational examples are provided throughout. The authors discuss tangles and operator invariants in Chapters 7 and 8, using the background material from earlier chapters. This progresses naturally to a discussion of the ribbon Hopf algebras and ends with the construction of the Reshetikhin-Turaev invariants. The third section of the book introduces chord diagrams and Vassiliev invariants. Jacobi diagrams and the STU relation are covered with computational examples and exercises. The section concludes with the application of Lie Algebra weight systems to Jacobi diagrams. In the final section of the book, the authors introduce the Kontsevich integral and demonstrate that it is a universal object for the Vassiliev invariants and the quantum invariants.
scientific article; zbMATH DE number 6999631